Mesh Quality Metrics Research Articles

Performance-Based Analysis of Mesh Smoothing Methods Combining Topology Optimization and Additive Manufacturing

Most design-to-manufacturing frameworks combining topology optimization (TO) and additive manufacturing (AM) integrate mesh smoothing methods as post-processing techniques to remove discrete irregularities of optimized topologies. Notably, a design framework is proposed incorporating all the CAD development stages within the design phase providing smooth and ready-to-print topologies. The Laplacian-based smoothing algorithms have demonstrated a high capacity in removing surface noise. This study focuses on investigating the smoothing capacity of both HC Laplacian and Taubin methods using mesh quality metrics to assess on their performance in terms of geometric preservation and volume shrinkage. Taubin method was found to produce high-quality smooth meshes with less volume shrinkage compared to HC Laplacian. The Taubin model exhibited an increase of 15.06% in mesh volume whereas the HC Laplacian model had a volume shrinkage of 28.14%. Additionally, finite element analyses of the three-point bending test using ANSYS is set to measure the flexural stiffness of an optimized MBB beam under both HC Laplacian and Taubin smoothing methods. Overall, the flexural stiffness of Taubin is nearly two times the original model with a surplus of 46.91% whereas HC Laplacian exhibited a flexural stiffness that is less with 72.07% than the original model.

Block‐structured mesh generation from implicit geometries for cardiovascular applications

AbstractGeneration of suitable meshes for patient‐specific vascular domains remains a major obstacle in the context of numerical simulations relying on the finite element method, as well as many similar methods. Herein, we discuss a skeleton‐based structured meshing approach, revolving around block‐structures. We rely on an implicitly represented surface, either via a signed distance function, or a convolution surface. The mesh generation using the aforementioned concepts is performed through the proper utilisation of transfinite maps. Application‐wise, we consider healthy blood vessels, and extensions required to tackle aortas afflicted by aortic dissection. The high quality of the resulting meshes is affirmed through common mesh quality metrics. Our approach features large flexibility in terms of spatial resolution, as well as mesh order.

LibHip: An open-access hip joint model repository suitable for finite element method simulation

Background and objective: population-based finite element analysis of hip joints allows us to understand the effect of inter-subject variability on simulation results. Developing large subject-specific population models is challenging and requires extensive manual effort. Thus, the anatomical representations are often subjected to simplification. The discretized geometries do not guarantee conformity in shared interfaces, leading to complications in setting up simulations. Additionally, these models are not openly accessible, challenging reproducibility. Our work provides multiple subject-specific hip joint finite element models and a novel semi-automated modeling workflow.Methods: we reconstruct 11 healthy subject-specific models, including the sacrum, the paired pelvic bones, the paired proximal femurs, the paired hip joints, the paired sacroiliac joints, and the pubic symphysis. The bones are derived from CT scans, and the cartilages are generated from the bone geometries. We generate the whole complex’s volume mesh with conforming interfaces. Our models are evaluated using both mesh quality metrics and simulation experiments.Results: the geometry of all the models are inspected by our clinical expert and show high-quality discretization with accurate geometries. The simulations produce smooth stress patterns, and the variance among the subjects highlights the effect of inter-subject variability and asymmetry in the predicted results.Conclusions: our work is one of the largest model repositories with respect to the number of subjects and regions of interest in the hip joint area. Our detailed research data, including the clinical images, the segmentation label maps, the finite element models, and software tools, are openly accessible on GitHub and the link is provided in Moshfeghifar et al.(2022)[1]. Our aim is to empower clinical researchers to have free access to verified and reproducible models. In future work, we aim to add additional structures to our models.

A novel neural network approach for airfoil mesh quality evaluation

Evaluating mesh quality before solving is crucially important for error control in the numerical simulation of airfoils. Traditional mesh quality metrics are used to identify distorted mesh elements by analyzing their geometric shape information like angles and edges. However, these metrics fail to recognize numerical errors stemming from quality attributes such as improper mesh density or distribution. This deficiency has imposed a burden on intensive manual re-evaluation, which heavily increases the meshing cost. In this paper, we introduce deep neural networks to airfoil mesh quality evaluation to improve its automation and efficiency. Specifically, we propose a neural network-based evaluation model, MQNet, accompanied by the first large-scale mesh benchmark dataset, AirfoilSet. MQNet is trained on the dataset to learn the quality-related attributes including mesh orthogonality, smoothness, and density. Thereafter, the trained network can be used as a black box to automatically evaluate and output the overall quality of the airfoil mesh. To demonstrate the effectiveness and generalization behaviors of the proposed network, we establish an MQNet-based mesh generation workflow. Experimental results show that MQNet is small and accurate. It outperforms widely-used neural networks and can be a useful tool to guide the high-quality mesh generation process with minimal user intervention.

Interactive all-hex meshing via cuboid decomposition

Standard PolyCube-based hexahedral (hex) meshing methods aim to deform the input domain into an axis-aligned PolyCube volume with integer corners; if this deformation is bijective, then applying the inverse map to the voxelized PolyCube yields a valid hex mesh. A key challenge in these methods is to maintain the bijectivity of the PolyCube deformation, thus reducing the robustness of these algorithms. In this work, we present an interactive pipeline for hex meshing that sidesteps this challenge by using a new representation of PolyCubes as unions of cuboids. We begin by deforming the input tetrahedral mesh into a near-PolyCube domain whose faces are loosely aligned to the major axis directions. We then build a PolyCube by optimizing the layout of a set of cuboids with user guidance to closely fit the deformed domain. Finally, we construct an inversion-free pullback map from the voxelized PolyCube to the input domain while optimizing for mesh quality metrics. We allow extensive user control over each stage, such as editing the voxelized PolyCube, positioning surface vertices, and exploring the trade-off among competing quality metrics, while also providing automatic alternatives. We validate our method on over one hundred shapes, including models that are challenging for past PolyCube-based and frame-field-based methods. Our pipeline reliably produces hex meshes with quality on par with or better than state-of-the-art. We additionally conduct a user study with 21 participants in which the majority prefer hex meshes they make using our tool to the ones from automatic state-of-the-art methods. This demonstrates the need for intuitive interactive hex meshing tools where the user can dictate the priorities of their mesh.

Alternative remeshing techniques for large deformation analysis of geotechnical problems

Mesh distortion is one of the main drawbacks of the Finite Element method when employed to analyse problems with relatively large deformations. The entanglement of elements caused by large deformations in a finite element mesh may lead to spontaneous termination of the analysis or inaccurate results. The Arbitrary Lagrangian-Eulerian method is a robust numerical technique widely used for problems involving large deformations. In this study four remeshing techniques including the Spring Analogy method, the Spring Analogy Elastic Hardening method, the Elastic Analys method and the Radial Basis Functions method are presented for analysing geotechnical problems within the framework of the Arbitrary Lagrangian-Eulerian strategy. The performance of these methods in reducing mesh distortion is comprehensively studied. In addition, several mesh quality metrics specifically designed for the 6-noded triangular elements are used to assess these mesh refinement methods. The performance of these mesh refienemnt techniques is investigated by solving a few geotechnical problems including the soil resistance under a rigid strip footing subjected to large deformations, the soil response due to loading on the edge of a vertical cut, dynamic penetration of a free-falling penetrometer into a layer of soil, and dynamic indentation of a sphere into a soil layer.

Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions

The mesh deformation method based on radial basis functions is widely used in computational fluid dynamics (CFD) simulation with a moving boundary. The traditional method for generating CFD mesh quality metrics called mesh-based metrics use the information of deformed mesh with specific element node coordinates and a connectivity relationship. This paper develops a new mesh quality, metric evaluating method based on the mapping process between the initial and deformed mesh, which is named mapping-based metrics. Mapping-based metrics are evaluated based on the conception of the deformation principle in continuum mechanics. This method provides a new point for mesh quality evaluation without requirements of deformed mesh coordinates and element connectivity information. Three test cases show that, comparing with indirectly solving by a geometrical method, mapping-based metrics accurately reveal the changes of the angle and area over the whole deformed domain. Additionally, the mapping-based metrics give high applicability to the quality of deformed mesh compared to mesh-based metrics. The quality evaluation method for CFD mesh proposed in this paper is effective.

Mesh quality metrics for isogeometric Bernstein–Bézier discretizations

High-order finite element methods harbor the potential to deliver improved accuracy per degree of freedom versus low-order methods. Their success, however, hinges upon the use of a curvilinear mesh of not only sufficiently high accuracy but also sufficiently high quality. In this paper, theoretical results are presented quantifying the impact of mesh parameterization on the accuracy of a high-order finite element approximation, and a formal definition of shape regularity is introduced for curvilinear meshes based on these results. This formal definition of shape regularity in turn inspires a new set of quality metrics for curvilinear finite elements. Computable bounds are established for these quality metrics using the Bernstein–Bézier form, and a new curvilinear mesh optimization procedure is proposed based on these bounds. Numerical results confirming the importance of shape regularity in the context of high-order finite element methods are presented, and numerical results demonstrating the promise of the proposed curvilinear mesh optimization procedure are also provided. The theoretical results in this paper apply to any piecewise-polynomial or piecewise-rational finite element method posed on a mesh of polynomial or rational mapped simplices and hypercubes. As such, they apply not only to classical continuous Galerkin finite element methods but also to discontinuous Galerkin finite element methods and even isogeometric methods based on NURBS, T-splines, or hierarchical B-splines.

Automatic triangular and triangular-prism mesh generation for overland and subsurface water flow simulations

In this paper, an automatic mesh generator is developed to simulate density-dependent water flow and transport problems in watershed systems. The two-dimensional (2D) triangular mesh is generated by global Delaunay triangulation, and the three-dimensional (3D) triangular-prism mesh is generated by vertical stretch. The implementation procedures of mesh generation are adjusted according to the interaction options of one-dimensional (1D) river, 2D overland, and 3D subsurface flows. An improved boundary point generation algorithm is proposed to maintain appropriate correspondence of points and edges according to the discrete patterns of river reaches and dead ends. Overlapped nodes are generated on zero-width river reaches and zero-width junctions without storage to construct the numerical models for 2D overland simulations. An additional triangular mesh generation method is proposed to create additional triangles for filling the empty water zones of finite-width river reaches, junctions with storage, finite-width ponds, lakes, and dead ends. The boundary loops bounding each water area are identified correctly, and the additional grids are created compatibly, aiming at the finite-depth and zero-depth patterns of storage zones. The computation equations of relevant parameters used for 3D stretch from triangles to triangular-prisms are built, and the 1D/2D/3D correspondence on river reaches, junctions, ponds, lakes, dead ends, and control structures is established. Finally, practical examples for discretizing real-world water areas are provided to demonstrate the robustness and effectiveness of our developed mesh generator by using the skewness as the mesh quality metric.

Improved Simultaneous Mesh Untangling and Quality Improvement Methods of 2D Triangular Meshes

Few inverted or poor quality elements can ruin both the accuracy and efficiency of numerical solutions. This paper proposes improved mesh untangling and quality improvement methods using the modified mesh quality metric for 2D triangular meshes. The new mesh quality metric ensures highly inverted elements assign higher penalty when the signed area of the element is negative, and also produces more accurate signed areas for valid elements. Therefore, the proposed methods can quickly remove inverted elements and also improve worst element qualities compared with the existing methods. We also combine Delaunay-based edge flip with optimization-based mesh untangling and quality improvement methods to further improve worst element qualities. Numerical results show that the proposed methods yield better quality meshes for improvement of the worst quality and are faster than existing methods.

Comparison of Meshing Strategies in THR Finite Element Modelling.

In biomechanics and orthopedics, finite element modelling allows simulating complex problems, and in the last few years, it has been widely used in many applications, also in the field of biomechanics and biotribology. As is known, one crucial point of FEM (finite element model) is the discretization of the physical domain, and this procedure is called meshing. A well-designed mesh is necessary in order to achieve accurate results with an acceptable computational effort. The aim of this work is to test a finite element model to simulate the dry frictionless contact conditions of a hip joint prosthesis (a femoral head against an acetabular cup) in a soft bearing configuration by comparing the performances of 12 common meshing strategies. In the simulations, total deformation of the internal surface of the cup, contact pressure, and the equivalent von Mises stress are evaluated by using loads and kinematic conditions during a typical gait, obtained from a previous work using a musculoskeletal multibody model. Moreover, accounting for appropriate mesh quality metrics, the results are discussed, underlining the best choice we identified after the large amount of numerical simulations performed.

Indicators for the Quality Assessment of the Grid Resolution

The choice of the grid resolution in thermo-fluid dynamics simulations is to be made as a balance between the quality of the results and the computational effort. When quick results are needed, coarse resolutions are often used, but in some cases this could be detrimental for the reliability of the numerical results. However, it is well known that reducing the size of the mesh affects dramatically the overall simulation time.Several approaches can be used to ensure that the mesh size is appropriate to reproduce the fire and smoke dynamics for a given scenario with suitable computational effort and make reliable predictions.The first and most used approach consists of limiting the ratio between the characteristic fire diameter D* and the grid size dx inside a range based on a series of experimental tests reported in literature. This approach is generally used as a first step in order to define a tentative grid resolution for the given model.Other approaches consist of checking if some indicators are included in a literature-based range once the simulation has run. For example, the dimensionless parameter y+ is a measure of how well the near-wall field is solved and can be effectively useful when the roughness of the walls affects the overall flow field. The measure of turbulence resolution (MTR), instead, is a quantity related to the resolved i.e. calculated turbulent kinetic energy and gives an idea of how detailed the turbulence effects are accounted in the model.These indicators are certainly a significant source of information regarding the quality of the model as well as the need of local refinement. However, they are not a guarantee for the achievement of convergent results of the quantity of interest (e.g. temperature, visibility, FED) which from the engineering point view are as valuable as the quality assessment of the model.In the paper, the traditional indicators are reviewed from a theoretical point of view and the Pearson’s coefficient is introduced as an attempt to add other statistical correlation principles. The purpose is to couple mesh quality metrics with a global convergence check of the quantity of interest of the fire safety analysis, driven also by aspects of conservativeness and computational cost. An application to a case study is also provided and shows pros and cons of the indicators.

A novel spatial pooling method for 3D mesh quality assessment based on percentile weighting strategy

Most of existing mesh quality assessment metrics consist of a similar two-stage computation process: first constructing a quality map by comparing the local regions between reference mesh and distorted mesh, then adopting a spatial pooling method to generate the overall quality score. In this paper, we propose a novel spatial pooling method for 3D mesh quality assessment based on percentile weighting strategy. We assign more weight to the severely distorted regions of the mesh at the pooling stage and extend the percentile weighting method by incorporating surface area at the pooling stage. Our analysis indicates that the percentile weighting method has a strong capability to emphasize the local regions with severe distortion of the mesh. We develop a mesh quality metric by pooling the local distances generated by the Tensor-based Perceptual Distance Measure metric with our spatial pooling method. We investigate the influence of the parameters of percentile weighting on the performance, and determine the optimal parameters and unified parameters through empirical tests on three publicly available databases. Experimental results demonstrate the effectiveness of our spatial pooling method and the superiority of our metric over state-of-the-art metrics.

(Re-) Meshing using interpolative mapping and control point optimization

Abstract This work proposes a simple and fast approach for re-meshing the surfaces of smooth-featured geometries prior to CFD analysis. The aim is to improve mesh quality and thus the convergence and accuracy of the CFD analysis. The method is based on constructing an interpolant based on the geometry shape and then mapping a regular rectangular grid to the shape of the original geometry using that interpolant. Depending on the selected interpolation algorithm the process takes from less than a second to several minutes. The main interpolant discussed in this article is a Radial Basis Function with cubic spline basis, however other algorithms are also compared. The mesh can be optimized further using active (flexible) control points and optimization algorithms. A range of objective functions are discussed and demonstrated. The difference between re-interpolated and original meshes produces a metric function which is indicative of the mesh quality. It is shown that the method works for flat 2D surfaces, 3D surfaces and volumes. Highlights RBF Map is created between key points on an arbitrary shape and an ideal equivalent. The rest of the points are recreated using reverse transformation to the original. Control points are automatically selected based on k-mean clustering. Optimization of the control points position based on desired mesh quality metrics. The method is shown to work both on 2D and 3D shapes as well as 3D surfaces.

A 3D Mesh Quality Metric based on Features Fusion

3D mesh becomes a common tool used in several computer vision applications. The performances of these applications depend highly on its quality. In order to quantify it, several methods have been proposed in the literature. In this paper, we propose a 3D Mesh Quality Measure based on the fusion of some selected features. The goal is here to take into account the advantages of these features and thus improve the global performance. The selected features are here some 3D mesh quality metrics and a geometric attribute. The fusion step has been realized using a Support Vector Regression (SVR) model. The 3D Mesh General database has been used to evaluate our method. The obtained results, in terms of correlation with the subjective judgments, show the relevance of the proposed framework.

A 3D Constrained Optimization Smoother to Post-process Quadrilateral Meshes for Body-in-white

Finite element analyses of 3D quadrilateral meshes for automotive body-in-white panels have stringent mesh quality requirements. Several mesh quality metrics, namely element included angles, minimum Jacobian determinant, skew, taper, warp, aspect ratio, minimum element length etc. need to be within acceptable limits. No constitutive relations exist that can tie all these parameters to a single metric that mesh post-processing can target. In the paper presented, a 3D optimization smoothing algorithm is proposed based on element included angles with the constraints of a minimum edge length and geometry fidelity envelope. A complex cost-function is set up for each element based on included element angle at the element corners. Element angle perturbation methods are devised to exercise local control on included angles of quadrilateral and mixed meshes. A minimization principle is worked out to reduce the cost function to an acceptable limit. Goal proximity is defined by acceptable error norms and ranges. Mesh nodes are repositioned iteratively but bound by a geometry fidelity envelope apart from the minimum element edge length constraint. Striking improvement in mesh quality statistics is reported with reasonably monotonic solution convergence patterns.

High quality surface reconstruction in radiotherapy: Cross-sectional contours to 3D mesh using wavelets.

A novel approach for three-dimensional (3D) surface reconstruction of anatomical structures in radiotherapy (RT) is presented. This is obtained from manual cross-sectional contours by combining both image voxel segmentation processing and implicit surface streaming methods using wavelets. 3D meshes reconstructed with the proposed approach are compared to those obtained from traditional triangulation algorithm. Qualitative and quantitative evaluations are performed in terms of mesh quality metrics. Differences in smoothness, detail and accuracy are observed in the comparison, considering three different anatomical districts and several organs at risk in radiotherapy. Overall best performances were recorded for the proposed approach, regardless the complexity of the anatomical structure. This demonstrates the efficacy of the proposed approach for the 3D surface reconstruction in radiotherapy and allows for further specific image analyses using real biomedical data.

An approach for patient-specific multi-domain vascular mesh generation featuring spatially varying wall thickness modeling

In this work, we present a computationally efficient image-derived volume mesh generation approach for vasculatures that implements spatially varying patient-specific wall thickness with a novel inward extrusion of the wall surface mesh. Multi-domain vascular meshes with arbitrary numbers, locations, and patterns of both iliac bifurcations and thrombi can be obtained without the need to specify features or landmark points as input. In addition, the mesh output is coordinate-frame independent and independent of the image grid resolution with high dimensional accuracy and mesh quality, devoid of errors typically found in off-the-shelf image-based model generation workflows. The absence of deformable template models or Cartesian grid-based methods enables the present approach to be sufficiently robust to handle aneurysmatic geometries with highly irregular shapes, arterial branches nearly parallel to the image plane, and variable wall thickness. The assessment of the methodology was based on i) estimation of the surface reconstruction accuracy, ii) validation of the output mesh using an aneurysm phantom, and iii) benchmarking the volume mesh quality against other frameworks. For the phantom image dataset (pixel size 0.105mm; slice spacing 0.7mm; and mean wall thickness 1.401±0.120mm), the average wall thickness in the mesh was 1.459±0.123mm. The absolute error in average wall thickness was 0.060±0.036mm, or about 8.6% of the largest image grid spacing (0.7mm) and 4.36% of the actual mean wall thickness. Mesh quality metrics and the ability to reproduce regional variations of wall thickness were found superior to similar alternative frameworks.

A priori mesh quality metrics for three-dimensional hybrid grids

Use of general hybrid grids to attain complex-geometry field simulations poses a challenge on estimation of their quality. Apart from the typical problems of non-uniformity and non-orthogonality, the change in element topology is an extra issue to address. The present work derives and evaluates an a priori mesh quality indicator for structured, unstructured, as well as hybrid grids consisting of hexahedra, prisms, tetrahedra, and pyramids. Emphasis is placed on deriving a direct relation between the quality measure and mesh distortion. The work is based on use of the Finite Volume discretization for evaluation of first order spatial derivatives. The analytic form of the truncation error is derived and applied to elementary types of mesh distortion including typical hybrid grid interfaces. The corresponding analytic expressions provide relations between the truncation error and the degree of stretching, skewness, shearing, torsion, expansion, as well as the type of grid interface.

Quality Metrics for High Order Meshes: Analysis of the Mechanical Simulation of the Heart Beat

The quality of a computational mesh is an important characteristic for stable and accurate simulations. Quality depends on the regularity of the initial mesh, and in mechanical simulations it evolves in time, with deformations causing changes in volume and distortion of mesh elements. Mesh quality metrics are therefore relevant for both mesh personalization and the monitoring of the simulation process. This work evaluates the significance, in meshes with high order interpolation, of four quality metrics described in the literature, applying them to analyse the stability of the simulation of the heart beat. It also investigates how image registration and mesh warping parameters affect the quality and stability of meshes. Jacobian-based metrics outperformed or matched the results of coarse geometrical metrics of aspect ratio or orthogonality, although they are more expensive computationally. The stability of simulations of a complete heart cycle was best predicted with a specificity of 61%, sensitivity of 85%, and only nominal differences were found changing the intra-element and per-element combination of quality values. A compromise between fitting accuracy and mesh stability and quality was found. Generic geometrical quality metrics have a limited success predicting stability, and an analysis of the simulation problem may be required for an optimal definition of quality.