Initial Mesh Research Articles

Flow2Mesh: A flow-guided data-driven mesh adaptation framework

Mesh adaptation is crucial in numerical simulation, providing optimal resource allocation for accurately capturing physical phenomena. However, when applied to Computational Fluid Dynamics (CFD) problems with complex multi-scale properties, existing adaptation methods face huge challenges due to the high computational cost of solving auxiliary partial differential equations (PDEs) and the difficulty in aligning the flow features with mesh geometric features. In this work, an end-to-end data-driven mesh adaptation framework, Flow2Mesh, is proposed to address these challenges by adopting a hybrid modeling strategy to construct the mapping from pixelated flow-fields to graph-based meshes. It achieves a rapid and accurate one-step mesh adaptation via a perceptual feature network (PFN) and a mesh movement network (MMN). PFN extracts the global perceptual features from flow-fields to enhance flow feature representation and mesh resolution independence. In MMN, these features are utilized to deform the initial mesh to a topology-invariant adaptive mesh by a proposed physically driven mesh convolutional network. It considers the inherent mesh geometric information for efficient node feature aggregation and alignment of mesh density with a flow-field structure. To generate high-quality adaptive meshes, various mesh-related losses are designed to regularize the mesh movement and alleviate the mesh tangling. Experiments in CFD scenarios demonstrate the generalization of our model to different design parameters and mesh configurations. It takes three orders of magnitude less time to generate similar meshes than the PDE-based method. The results exhibit the potential of Flow2Mesh to be a flexible and reliable tool for rapid mesh adaptation in scientific and industrial fields.

3D Reconstruction of Wheat Plants by Integrating Point Cloud Data and Virtual Design Optimization

The morphology and structure of wheat plants are intricate, containing numerous tillers, rich details, and significant cross-obscuration. Methods of effectively reconstructing three-dimensional (3D) models of wheat plants that reflects the varietal architectural differences using measured data is challenging in plant phenomics and functional–structural plant models. This paper proposes a 3D reconstruction technique for wheat plants that integrates point cloud data and virtual design optimization. The approach extracted single stem number, growth position, length, and inclination angle from the point cloud data of a wheat plant. It then built an initial 3D mesh model of the plant by integrating a wheat 3D phytomer template database with variety resolution. Diverse 3D wheat plant models were subsequently virtually designed by iteratively modifying the leaf azimuth, based on the initial model. Using the 3D point cloud of the plant as the overall constraint and setting the minimum Chamfer distance between the point cloud and the mesh model as the optimization objective, we obtained the optimal 3D model as the reconstruction result of the plant through continuous iterative calculation. The method was validated using 27 winter wheat plants, with nine varieties and three replicates each. The R2 values between the measured data and the reconstructed plants were 0.80, 0.73, 0.90, and 0.69 for plant height, crown width, plant leaf area, and coverage, respectively. Additionally, the Normalized Root Mean Squared Errors (NRMSEs) were 0.10, 0.12, 0.08, and 0.17, respectively. The Mean Absolute Percentage Errors (MAPEs) used to investigate the vertical spatial distribution between the reconstructed 3D models and the point clouds of the plants ranged from 4.95% to 17.90%. These results demonstrate that the reconstructed 3D model exhibits satisfactory consistency with the measured data, including plant phenotype and vertical spatial distribution, and accurately reflects the characteristics of plant architecture and spatial distribution for the utilized wheat cultivars. This method provides technical support for research on wheat plant phenotyping and functional–structural analysis.

An adaptive extended finite element based crack propagation analysis method

In this paper, a method of crack propagation analysis based on adaptive extension finite element is proposed. This method combines adaptive mesh reconstruction technology with extended finite element method (XFEM). Firstly, the model of engineering structure is discretized with the help of mesh generation software, and the initial mesh is divided. Secondly, Construction of XFEM model, and the tip of crack strengthening function is introduced to describe the physical field properties of the crack tip. The integral equation is solved to obtain the crack tip parameters. Then, the adaptive mesh reconstruction technology is built to refine the mesh of crack tip area through the error estimation of crack tip. Finally, the SIFs at the crack tip were calculated using the interaction integral, and the path direction of crack growth was determined using the maximum circumferential tensile stress criterion. Thus, the propagation path can be well traced.

A quasi-meshfree method for constructing boundary-aware reproducing bases on geometrically complex domains using manifold geodesics

In many applications, physical domains are geometrically complex making it challenging to perform coarse-scale approximation. A defeaturing process is often used to simplify the domain in preparation for approximation and analysis at the coarse scale. Herein, a methodology is presented for constructing a coarse-scale reproducing basis on geometrically complex domains given an initial fine-scale mesh of the fully featured domain. The initial fine-scale mesh can be of poor quality and extremely refined. The construction of the basis functions begins with a coarse-scale covering of the domain and generation of weighting functions with local support. Manifold geodesics are used to define distances within the local support for general applicability to non-convex domains. Conventional moving least squares is used to construct the coarse-scale reproducing basis. Applications in quasi-interpolation and linear elasticity are presented.

Parallel and automatic mesh sizing field generation for complicated CAD models

PurposeThe purpose of this paper is to present an automatic approach for mesh sizing field generation of complicated computer-aided design (CAD) models.Design/methodology/approachIn this paper, the authors present an automatic approach for mesh sizing field generation. First, a source point extraction algorithm is applied to capture curvature and proximity features of CAD models. Second, according to the distribution of feature source points, an octree background mesh is constructed for storing element size value. Third, mesh size value on each node of background mesh is calculated by interpolating the local feature size of the nearby source points, and then, an initial mesh sizing field is obtained. Finally, a theoretically guaranteed smoothing algorithm is developed to restrict the gradient of the mesh sizing field.FindingsTo achieve high performance, the proposed approach has been implemented in multithreaded parallel using OpenMP. Numerical results demonstrate that the proposed approach is remarkably efficient to construct reasonable mesh sizing field for complicated CAD models and applicable for generating geometrically adaptive triangle/tetrahedral meshes. Moreover, since the mesh sizing field is defined on an octree background mesh, high-efficiency query of local size value could be achieved in the following mesh generation procedure.Originality/valueHow to determine a reasonable mesh size for complicated CAD models is often a bottleneck of mesh generation. For the complicated models with thousands or even ten thousands of geometric entities, it is time-consuming to construct an appropriate mesh sizing field for generating high-quality mesh. A parallel algorithm of mesh sizing field generation with low computational complexity is presented in this paper, and its usability and efficiency have been verified.

An Efficient Parallel Iterative Solver for Controlled-source Electromagnetic 3-D Adaptive Forward Modeling in General Anisotropic Media

Controlled-source electromagnetic (CSEM) surveys are moving increasingly toward realistic and complicated scenarios where the survey region may contain undulated topography, complex geometries, and electrical anisotropy media. In this paper, the adaptive finite element numerical method is employed to discrete the total electric field equation, which can provide precise electromagnetic (EM) responses even with a coarse initial mesh. The unstructured tetrahedral grid is employed to effectively address arbitrary irregular geometry and mountain terrain. Then, the flexible generalized minimum residual solver (FGMRES), auxiliary-space Maxwell pre-conditioner, and grid division technique were used to solve the large-scale linear system of equations, which can stably solve ill-conditioned problems using fewer computing resources. Finally, we conducted a numerical experiment via our newly proposed forward modeling scheme on a synthetic model with multi-contrast electrical anisotropy. It validated that accurate EM fields could be obtained against the semi-analytic solutions, and this iterative solver has good robustness for various anisotropic media. As a result, we have developed a state-of-the-art 3D CSEM anisotropic forward modeling engine, which can quickly and accurately deal with large-scale and complex geo-models.

R-Adaptive deep learning method for solving partial differential equations

We introduce a Deep Neural Network (DNN) method for solving Partial Differential Equations (PDEs) that simultaneously: (a) constructs an optimal r-adapted mesh, i.e., given an initial mesh, it provides optimal node locations, and (b) solves the PDE over the constructed r-adaptive mesh. The node locations are optimized over a set of 1D boundary nodes, and the corresponding 2D quadrilateral meshes are built using tensor product. The method supports the definition of fixed interfaces to create conforming meshes and allows nodes to jump across them, permitting topological variations. To numerically illustrate the performance of our novel r-adaptive deep learning method, we apply it in combination with other numerical methods including collocation, Least Squares, and the Deep Ritz method. We solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients. Results consistently show the outperformance of employing r-adaptivity, while in some cases the improvement is limited by the tensor-product structure of the mesh.

Numerical simulation of the extrusion process of viscoplastic materials using a radial point interpolation method

PurposeFused Filament Fabrication (FFF) is an extrusion-based manufacturing process using fused thermoplastics. Despite its low cost, the FFF is not extensively used in high-value industrial sectors mainly due to parts' anisotropy (related to the deposition strategy) and residual stresses (caused by successive heating cycles). Thus, this study aims to investigate the process improvement and the optimization of the printed parts.Design/methodology/approachIn this work, a meshless technique – the Radial Point Interpolation Method (RPIM) – is used to numerically simulate the viscoplastic extrusion process – the initial phase of the FFF. Unlike the FEM, in meshless methods, there is no pre-established relationship between the nodes so the nodal mesh will not face mesh distortions and the discretization can easily be modified by adding or removing nodes from the initial nodal mesh. The accuracy of the obtained results highlights the importance of using meshless techniques in this field.FindingsMeshless methods show particular relevance in this topic since the nodes can be distributed to match the layer-by-layer growing condition of the printing process.Originality/valueUsing the flow formulation combined with the heat transfer formulation presented here for the first time within an in-house RPIM code, an algorithm is proposed, implemented and validated for benchmark examples.

The investigation of Sn corrosion on Mo meshes under the irradiation of high-flux hydrogen plasma

Liquid-metal divertor consisting of liquid Sn and Mo meshes would face high-flux plasma environments during service. In this work, the Sn corrosion behaviors on initial and pre-irradiated Mo meshes under hydrogen plasma irradiation were investigated to evaluate the reliability of the porous structure. According to the analysis and computation of XRD spectrum, the sub-grain size of the initial meshes would continuously grow during the corrosion process. The SEM images showed that after corrosion, the appearance and disappearance of prismatic Mo grains occurred alternately with the increase of plasma fluence. The thicknesses of corroded wires in the longitudinal and transverse directions displayed a similar repeatable change. In comparison to initial meshes, surface sub-grain growth and damage induced by the pre-irradiation treatment would result in differences in the corrosion process under hydrogen plasma irradiation. The pre-irradiated surface Mo grains with obvious grain boundaries were first dissolved to pebble-like Mo particles by liquid Sn and then stripped from the surface after corrosion. Sub-grain size was reduced from 72.5 nm to 41.1 nm at the fluence of 1.07 × 1026 ions/m2, which was inverse to the increasing phenomenon of the initial sample under same corrosion conditions. Additionally, the corrosion phenomenon, i.e., the presence and subsequent disappearance of prismatic Mo grains, occurred similarly on the surface of pre-irradiated samples. Based on the analysis of initial and pre-irradiated surface morphologies and wire thickness, Mo grains might go through an onion-peeling-like cycle of growth and fracture during the corrosion process. Besides, the wire thicknesses of all pre-irradiated Mo meshes were slightly less than that of the initial samples under the same corrosion conditions. Somehow, the comparative result reflected that the pre-irradiated samples were corroded more severely. Thus, the pre-irradiation of hydrogen plasma in this work would weaken corrosion resistance of Mo meshes in liquid Sn.

Heat Production Capacity Simulation and Parameter Sensitivity Analysis in the Process of Thermal Reservoir Development

The development of a geothermal system involves changes in the temperature field (T), seepage field (H), stress field (M), and chemical field (C) and the influence among them and injecting the heat extraction working fluid into the injection well that flows (migrating) through natural fractures and exchanges heat with the geothermal high-temperature rock. At the same time, the injection of low-temperature working fluid will induce thermal stress, resulting in changes in the reservoir temperature field and stress field. To study the influence factors and influence degree of heat production performance and mining life under multi-field coupling in the process of thermal reservoir development, based on THMC multi-field coupling numerical simulation software, this paper deeply studies the control differential equations and boundary coupling conditions of rock mass (fracture) deformation, seepage, heat exchange, the chemical reaction, and other processes based on the numerical solution method of the discrete fracture network model, simulating heat production capacity during the deep geothermal resource extraction process. The reservoir geological model analysis and generalization, parameter setting, boundary conditions, initial condition settings, mesh generation, and other steps were carried out in turn. Two different heat extraction working fluids, water, and CO2 were selected for numerical simulation in the mining process. The changes in the thermal reservoir temperature, net heat extraction rate, and SiO2 concentration during the thirty years of systematic mining were compared. The results show that CO2 has a better heat extraction effect. Finally, the reservoir thermal conductivity, heat capacity, well spacing, injection temperature, fracture spacing, fracture permeability, fracture number, fracture length, and other parameters were set, respectively. The parameter variation range was set, and the parameter sensitivity analysis was carried out. The numerical simulation results show that the engineering production conditions (injection temperature, well spacing) have little effect on the thermal efficiency and mining life, and the properties of fractures (fracture permeability, fracture number, fracture length) have a great influence.

Evaluation of Mechanical Stress Distribution Across the Calcaneus following Calcaneal Autograft: A Finite Element Analysis

Category: Other Introduction/Purpose: The purpose of this study was to utilize finite element analysis to evaluate progressive bone removal from the calcaneus at various angles and its effect on the mechanical stress distribution at the calcaneus, and potential increase in fracture risk. Methods: A calcaneus CT scan was imported into Autodesk Fusion360 (Autodesk, v.2.0.14569). After rendering the initial mesh body using tetrahedral elements, a copy of the calcaneus was made. A simulated force equal to the average weight of person (876N) was applied to the middle and posterior talus facets while the calcaneus was oriented in the heel strike position during gait, with constraints applied around the heel. The initial harvest site was modelled as a 6mm diameter 30mm deep cylinder. From this center point, the diameter of the harvest site was increased to 8, and 10mm, with the depth being adjusted such that there was no risk of breaching the cortex on the opposite side. For each iteration, the safety factor, stress (MPa), and strain were recorded. This was repeated at an angle of 90°, 45° and a combination of 45° and 90°. Results: The 90° model demonstrated that as the diameter of the harvest site increased, there was a reciprocal increase in stress across the calcaneus. The number of cycles to failure (fracture) for all diameters was 10^6-10^7. The 45° model demonstrated that as the diameter of the harvest site increased, there was a reciprocal increase in stress applied across the calcaneus with a corresponding decrease in stress at the harvest site. The number of cycles to failure increased from 10^3 at 6mm to 10^4-10^5 at 10mm. The combination of the 45°-90° model demonstrated that as the diameter of the harvest site increased, there was a reciprocal increase in stress applied across the calcaneus. The cycles to failures decreased as the diameter of the harvest site increased. Conclusion: This finite element analysis demonstrated that an increase in the diameter of the harvest site led to an increase in stress along the calcaneus. There was variation in the correlation between size of the diameter and the number of cycles to failure depending on the angle of the harvest. The greatest amount of stress applied at the calcaneus was with the 45°-90° model. It was determined that harvesting at an angle of 45° at 6mm produced the lowest number of cycles to failure.

Concurrent optimization method of principal stress orientation interpolated continuous fiber angle (PSO-CFAO) and structural topology

Continuous fiber-reinforced polymers (CFRPs) have been widely applied in aerospace and other fields due to their excellent mechanical properties, which highly depends on the material distribution and fiber orientations. The designability of CFRP structures and fiber distributions provides an opportunity for achieving better physical properties through optimization. However, local optima and the dependence of initial fiber angle variables make the concurrent optimization of fiber orientation and topology a challenging problem. In this paper, the principal stress orientation interpolated continuous fiber angle optimization (PSO-CFAO) method combined with the independent continuous mapping (ICM) method is proposed to realize the design of CFRP structures with a clear macroscopic topology and microscopic fiber distribution. A sigmoid function is applied to interpolate the fiber angle variables by the principal stress orientation. The fiber angle variables are modified and a continuous fiber design is obtained during the iteration process, which reduces the possibility of a local optimum. Several examples are provided to prove the effectiveness and stability of the proposed method, and the expected results are acquired for different initial fiber angles, material parameters and mesh densities. The proposed method provides guidance for the design of CFRP structures and the planning of fiber laying paths.

Fluid‐structure interaction with fully coupled mesh generation

AbstractThe modeling and simulation of fluid‐structure interaction (FSI) involve the two‐way coupling of a fluid flow and a deforming structure. The fluid exerts a load on the structure, the structure deforms, and, hence, the fluid flow changes, which leads to an altered loading on the structure. To accomplish this coupling, the fluid domain has to be updated after each time step, leading to a change in the fluid mesh. Various mesh update methods have been presented where an initial mesh is generated and then updated after each time step. Each method comes with different levels of complexity and has its own advantages and disadvantages. Herein, FSI simulations with fully integrated mesh generation, rather than updates, are proposed. Instead of updating the mesh at each time step, a new mesh for the fluid domain is generated based on the deformation of the solid domain. An advanced structured meshing algorithm, based on a block structure, enables this integrated mesh generation approach. An initial set of coarse‐scale conforming blocks is generated by the user, representing the topology and rough initial position of the solid and the fluid. Further information is the exact geometry at the boundary and the deformed position of the fluid‐structure interface, plus grading information to obtain a high‐quality fluid mesh. Transfinite maps are used to generate elements inside the blocks with any desired resolution and order.

An [formula omitted]-adaptive discontinuous Galerkin method for phase field fracture

The phase field method is becoming the de facto choice for the numerical analysis of complex problems that involve multiple initiating, propagating, interacting, branching and merging fractures. However, within the context of finite element modelling, the method requires a fine mesh in regions where fractures will propagate, in order to capture sharp variations in the phase field representing the fractured/damaged regions. This means that the method can become computationally expensive when the fracture propagation paths are not known a priori. This paper presents a 2D hp-adaptive discontinuous Galerkin finite element method for phase field fracture that includes a posteriori error estimators for both the elasticity and phase field equations, which drive mesh adaptivity for static and propagating fractures. This combination means that it is possible to be reliably and efficiently solve phase field fracture problems with arbitrary initial meshes, irrespective of the initial geometry or loading conditions. This ability is demonstrated on several example problems, which are solved using a light-BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasi-Newton algorithm. The examples highlight the importance of driving mesh adaptivity using both the elasticity and phase field errors for physically meaningful, yet computationally tractable, results. They also reveal the importance of including p-refinement, which is typically not included in existing phase field literature. The above features provide a powerful and general tool for modelling fracture propagation with controlled errors and degree-of-freedom optimised meshes.

Iterated Residual Graph Convolutional Neural Network for Personalized Three-Dimensional Reconstruction of Left Myocardium from Cardiac MR Images.

Three-dimensional reconstruction of the left myocardium is of great significance for the diagnosis and treatment of cardiac diseases. This paper proposes a personalized 3D reconstruction algorithm for the left myocardium using cardiac MR images by incorporating a residual graph convolutional neural network. The accuracy of the mesh, reconstructed using the model-based algorithm, is largely affected by the similarity between the target object and the average model. The initial triangular mesh is obtained directly from the segmentation result of the left myocardium. The mesh is then deformed using an iterated residual graph convolutional neural network. A vertex feature learning module is also built to assist the mesh deformation by adopting an encoder-decoder neural network to represent the skeleton of the left myocardium at different receptive fields. In this way, the shape and local relationships of the left myocardium are used to guide the mesh deformation. Qualitative and quantitative comparative experiments were conducted on cardiac MR images, and the results verified the rationale and competitiveness of the proposed method compared to related state-of-the-art approaches.

Hybrid Navier–Stokes–Direct Simulation Monte Carlo Automatic Mesh Optimization for Hypersonics

This paper describes the development of an unstructured hybrid finite element Navier–Stokes (NS)–direct simulation Monte Carlo (DSMC) framework for hypersonic flows. State-based coupling is employed and simulations of varying thermochemical complexity demonstrate the accuracy, robustness, and computational efficiency of the hybrid all-Mach algorithm. An automatic mesh optimization process using a posteriori error estimates based on the Hessian of the solution goes much further than traditional mesh adaptation processes by equidistributing the error estimator and producing a “single optimal hybrid mesh” with no increase in mesh size and with much higher accuracy. The DSMC region cells of the resulting optimal mesh are smaller than in NS regions and are sized to the local mean free path. Mesh optimization is also shown to greatly improve the quality of the hybrid interfaces from those of the initial mesh. Unstructured meshes are found to represent the hybrid interfaces smoothly, while structured meshes showcase a castellated pattern in the interfaces. The optimal hybrid meshes are found to be statistically similar to optimal full DSMC meshes, thus highlighting the solver independence of the optimizer. Such a coupled hybrid mesh optimization strategy can therefore tackle hypersonic flows with multiscale flow features at any degree of rarefaction.

Auto Mesh generation algorithm for the convex domain with the triangular elements

Mesh creation is one of the primary tasks in implementing the Finite Element Method (FEM) to solve two- and three-dimensional boundary value problems. However, the whole procedure becomes tedious and problematic when higher-order finite elements are employed to construct mesh and prepare corresponding element data. In this study, we strive to develop a versatile algorithm for discretizing the two-dimensional domain using linear, quadratic, and cubic triangular finite elements. The algorithm is developed based on n vertices (actual or more in number) that constitute the boundary of the domain, along with a computed (n+1)-th point such as the centroid. The algorithm, using this input, generates an initial mesh consisting of n triangular elements. Then, in every subsequent step, the algorithm increases the number of triangles in the mesh fourfold compared to the previous step. The algorithm we present here can employ (a) straight-sided (linear, quadratic, and cubic) triangular elements to generate the mesh for domains with polygonal boundaries and (b) both straight-sided and curved (quadratic and cubic) triangular elements with two straight sides and one curved side to generate meshes for domains with curved boundaries. Thus, the algorithm generates the desired meshes if the vertices are specified once at the initial step. Additionally, the inclusion of the mathematical expression of the curved boundary enables the algorithm to generate the fine mesh for the curved domain utilizing higher-order curved and straight-sided triangular elements. We also present a computer code in MATLAB incorporating the algorithm to create the mesh, prepare the element's data, and determine the element's connectivity.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 017- 035

Self-Adaptive Mesh Generator for Global Complex Roots and Poles Finding Algorithm

In any global method of searching for roots and poles, increasing the number of samples increases the chances of finding them precisely in a given area. However, the global complex roots and poles finding algorithm (GRPF) (as one of the few) has direct control over the accuracy of the results. In addition, this algorithm has a simple condition for finding all roots and poles in a given area: it only requires a sufficiently dense initial grid. However, in practice, this requirement may turn out to be very difficult to implement. For a complex and sophisticated function, the use of a regular high-density mesh may be ineffective or even impossible due to limited computational resources. In this article, a method for creating a self-adaptive initial mesh for this algorithm is presented. The proposed solution uses gradient calculation to identify areas that require mesh refinement, including areas where a zero and a pole are in close proximity. The adaptive mesh allows for faster and more accurate analysis of functions with a much smaller number of samples. As shown in the numerical examples, this approach reduces the number of function calls by several orders of magnitude, and thus significantly reduces the computational time.

Wada boundaries on a hyperbolic pair of pants

In this paper, the geodesics of an open multiply connected hyperbolic manifold are presented from the dynamical system point of view. The approach is completely numerical. Similar to the closed hyperbolic case there is a zero-measure set of periodic orbits. The difference is that now, most of the geodesics escape to infinity through one of the topologically distinct channels, and initial conditions are identified to which channel the orbit escapes. The initial condition mesh reveals basins of attraction to each channel. We verified that the basin boundary points are of the Wada type by two independent methods. We have also calculated the basin entropy of the system and verified that it depends on the opening angle of the escape channels. The manifold chosen is the pair of pants together with the leaves, so there are 3 distinct exit channels.

A Study on the Ultimate Strength and Failure Mode of Stiffened Panels

Different buckling failure modes of stiffened panels will result in differences in ultimate strengths. In this study, the benchmark model of the bottom of bulk carrier with appropriate initial imperfections, boundary conditions and mesh size are selected for a series of non-linear FE analyses. A failure mode discrimination method considering the change rule of ultimate strength and the stress and strain distribution of stiffened plates in ultimate limit state is proposed, and then the boundary function of the failure modes composed of four key parameters, λ, β, hw/tw, λe, is established. Based on this boundary function, the rapid identification of failure modes and the classification of ultimate strength under different failure modes can be realized. Furthermore, the ultimate strength formulas of stiffened panels for different failure modes are obtained by data analysis from many nonlinear finite element analyses.