High-order Mesh Research Articles

Minimal surface-guided higher-order mesh generation for CAD models

This paper presents a novel method for generating higher-order meshes for CAD surfaces by leveraging minimal surface theory to improve element shapes. We explore the concept of higher-order mesh distortion through deformation gradients and introduce an energy function designed to minimize the surface area of these meshes, providing a theoretical justification for its effectiveness in untangling. The process of mesh generation starts with segmenting CAD surfaces into linear elements, followed by the insertion of higher-order nodes within these elements. These nodes are then projected onto the CAD surface to form the initial higher-order elements. By optimizing energy functions related to minimal surfaces and the projection distances, we achieve high-quality, geometrically accurate higher-order surface meshes. Our method has been validated on complex geometries, showcasing its potential in creating effective higher-order meshes for industrial CAD models.

Evolutionary multi-objective high-order tetrahedral mesh optimization

High-order mesh optimization has many goals, such as improving smoothness, reducing approximation error, and improving mesh quality. The previous methods do not optimize these objectives together, resulting in suboptimal results. To this end, we propose a multi-objective optimization method for high-order meshes. Central to our algorithm is using the multi-objective genetic algorithm (MOGA) to adapt to the multiple optimization objectives. Specifically, we optimize each control point one by one, where the MOGA is applied. We demonstrate the feasibility and effectiveness of our method over various models. Compared to other state-of-the-art methods, our method achieves a favorable trade-off between multiple objectives.

High-order shape interpolation

We propose a simple yet effective method to interpolate high-order meshes. Given two manifold high-order triangular (or tetrahedral) meshes with identical connectivity, our goal is to generate a continuum of curved shapes with as little distortion as possible in the mapping from the source mesh to the interpolated mesh. Our algorithm contains two steps: (1) linearly blend the pullback metric of the identity mapping and the input mapping between two Bézier elements on a set of sampling points; (2) project the interpolated metric into the metric space between Bézier elements using the Newton method for nonlinear optimization. We demonstrate the feasibility and practicability of the method for high-order meshes through extensive experiments in both 2D and 3D.

A fast method for robust estimation of gearbox instantaneous speed

Under non-stationary condition, estimating the instantaneous frequency (IF) of shaft from vibration signal is an ideal way for gearbox fault monitoring without tachometer. Among them, the method based on harmonic phase demodulation has always occupied a dominant position. In the low signal-to-noise ratio (SNR) condition, the noisy harmonic components may cause greater IF estimation error. In this work, an improved approach for estimating the IF of gearbox reference shaft is proposed based on the short time Fourier transform spectrogram. The theoretical derivation proves that the high-order meshing harmonic can reduce the IF estimation error of gearbox reference shaft. By constructing the spectrogram weight factor, an improved maximum tracking algorithm is proposed to redefine the IF estimation range. On this basis, the robust IF estimation of high-order meshing harmonic with low SNR is realized. Finally, it is equivalent to the estimated IF of gearbox reference shaft. The proposed strategy is compared with the two advanced IF estimation methods. The civil aircraft engine dataset and the wind turbine gearbox dataset provided in the Safran contest demonstrate that the proposed strategy works. In the case of heavy load and large speed change, the proposed method has extremely low consumption time while maintains good estimation accuracy.

NekMesh: An open-source high-order mesh generation framework

High-order spectral element simulations are now becoming increasingly popular within the computational modelling community, as they offer the potential to deliver increased accuracy at reduced cost compared to traditional low-order codes. However, to support accurate, high-fidelity simulations in complex industrial applications, there is a need to generate curvilinear meshes which robustly and accurately conform to geometrical features. This is, at present, a key challenge within the mesh generation community, with only a few open-source tools able to generate curvilinear meshes for complex geometries. We present NekMesh: an open-source mesh generation package which is designed to enable the generation of valid, high-quality curvilinear meshes of complex, three-dimensional geometries for performing high-order simulations. We outline the software architecture adopted in NekMesh, which uses a pipeline of processing modules to provide a flexible, CAD-independent high-order mesh processing tool, capable of both generating meshes for a wide range of use cases, as well as post-processing linear meshes from a range of input formats for use with high-order simulations. A number of examples in various application areas are presented, with a particular emphasis on challenging aeronautical and fluid dynamics test cases. Program summaryProgram title: NekMesh (version 5.4.0)CPC Library link to program files:https://doi.org/10.17632/d82hjm4v6r.1Licensing provisions: MITProgramming language: C++External routines/libraries: Boost, TinyXML, OpenCASCADE, Triangle, TetGen, HDF5Nature of problem: NekMesh is a high-order mesh generation framework with the goal of providing a robust framework to automate the process of generating valid meshes for complex three-dimensional CAD geometries.Solution method: Energy minimisation, solid body models and other techniques based around high-order finite element methods.Additional comments including restrictions and unusual features: The supplied cases require a system with 16384 MB of RAM for the automotive example and 32768 MB of RAM for the variational optimisation example.

Defining metric-aware size-shape measures to validate and optimize curved high-order meshes

We define a regularized size-shape distortion (quality) measure for curved high-order elements on a Riemannian space. To this end, we measure the deviation of a given element, straight-sided or curved, from the stretching, alignment, and sizing determined by a target metric. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements on Riemannian spaces determined by constant and point-wise varying metrics. The examples illustrate that the distortion can be minimized to curve (deform) the elements of a given high-order (linear) mesh and try to match with curved (linear) elements the point-wise stretching, alignment, and sizing of a discrete target metric tensor. In addition, the resulting meshes simultaneously match the curved features of the target metric and boundary. Finally, to verify if the minimization of the metric-aware size-shape distortion leads to meshes approximating the target metric, we compute the Riemannian measures for the element edges, faces, and cells. The results show that, when compared to anisotropic straight-sided meshes, the Riemannian measures of the curved high-order mesh entities are closer to unit. Furthermore, the optimized meshes illustrate the potential of curved r-adaptation to improve the accuracy of a function representation.

A Globalized and Preconditioned Newton-CG Solver for Metric-Aware Curved High-Order Mesh Optimization

We present a specific-purpose globalized and preconditioned Newton-CG solver to minimize a metric-aware curved high-order mesh distortion. The solver is specially devised to optimize curved high-order meshes for high polynomial degrees with a target metric featuring non-uniform sizing, high stretching ratios, and curved alignment — exactly the features that stiffen the optimization problem. To this end, we consider two ingredients: a specific-purpose globalization and a specific-purpose Jacobi-iLDLT(0) preconditioning with varying accuracy and curvature tolerances (dynamic forcing terms) for the CG method. These improvements are critical in stiff problems because, without them, the large number of non-linear and linear iterations makes curved optimization impractical. First, to enhance the global convergence of the non-linear solver, the globalization strategy modifies Newton’s direction to a feasible step. In particular, our specific-purpose globalization strategy memorizes the length of the feasible step (step-length continuation) between the optimization iterations while ensuring sufficient decrease and progress. Second, to compute Newton’s direction in second-order optimization problems, we consider a conjugate-gradient iterative solver with specific-purpose preconditioning and dynamic forcing terms. To account for the metric stretching and alignment, the preconditioner uses specific orderings for the mesh nodes and the degrees of freedom. We also present a preconditioner switch between Jacobi and iLDLT(0) preconditioners to control the numerical ill-conditioning of the preconditioner. In addition, the dynamic forcing terms determine the required accuracy for the Newton direction approximation. Specifically, they control the residual tolerance and enforce sufficient positive curvature for the conjugate-gradients method. Finally, to analyze the performance of our method, the results compare the specific-purpose solver with standard optimization methods. For this, we measure the matrix–vector products indicating the solver computational cost and the line-search iterations indicating the total amount of objective function evaluations. When we combine the globalization and the linear solver ingredients, we conclude that the specific-purpose Newton-CG solver reduces the total number of matrix–vector products by one order of magnitude. Moreover, the number of non-linear and line-search iterations is mainly smaller but of similar magnitude.

High-Order Finite Element Second Moment Methods for Linear Transport

We present high-order, finite element–based Second Moment Methods (SMMs) for solving radiation transport problems in two spatial dimensions. We leverage the close connection between the Variable Eddington Factor (VEF) method and SMM to convert existing discretizations of the VEF moment system to discretizations of the SMM moment system. The moment discretizations are coupled to a high-order Discontinuous Galerkin discretization of the SN transport equations. We show that the resulting methods achieve high-order accuracy on high-order (curved) meshes, preserve the thick diffusion limit, and are effective on a challenging multimaterial problem both in outer fixed-point iterations and in inner preconditioned iterative solver iterations for the discrete moment systems. We also present parallel scaling results and provide direct comparisons to the VEF algorithms from which the SMM algorithms were derived.

An efficient implementation of compact third-order implicit reconstruction solver with a simple WBAP limiter for compressible flows on unstructured meshes

In this paper, we present the development of a third-order density-based solver within the OpenFOAM framework, tailored for handling compressible flows. The solver incorporates implicit variational reconstruction on three-dimensional unstructured meshes, as well as a novel technology coupling reconstruction and time integration for both steady and unsteady simulations. To address the challenge of achieving high-order accuracy for curved geometries, we introduce a new approach for curved wall boundary reconstruction, specifically designed for situations where high-order mesh information is not readily available in OpenFOAM. Furthermore, we propose a simple WBAP limiter capable of capturing shocks without necessitating the whole-domain successive limiting procedure. Numerical tests were conducted to assess the solver's performance. The results reveal that our established solver exhibits higher accuracy in smooth flow simulations, while maintaining an excellent balance between accuracy and robustness for problems involving strong shocks and other complex flow structures.

Effective evaluation of aerodynamic characteristics using subparametric finite element transformation for unmanned air vehicles at low Reynolds number

The current article delineates the importance of the aerodynamic characteristics of airfoils utilized in unmanned aerial vehicles (UAVs) and their evaluation. UAVs are excessively in demand for the major use of medical applications in the present situations where unexpected climatic changes occur. The increasing demand for UAVs in medical applications like healthcare, transporting medicines to remote areas of medical necessity, and surveillance of affected areas and disastrous sites having biological hazards. The prerequisite transportation of essential medical help for the individuals or groups of crucial areas such as those affected by floods, drought, or other natural calamities can be efficiently carried out by UAVs. There is currently a significant demand for airfoils that can enhance the lift and improve the efficiency of UAVs used in medical applications. The finite element meshing technique is valuable in accurately assessing the aerodynamic properties and selecting suitable airfoils for UAVs. In this regard, the NACA 0009 and NACA 6409 airfoils are frequently utilized, and their lift coefficient, drag coefficient, and lift-to-drag ratio have been evaluated at low Reynolds numbers. To ensure precise evaluation, an innovative subparametric higher-order meshing technique has been employed for these specific airfoils. It has shown a tremendously better result with the higher order meshing techniques.

Fast High-Order Mesh Correction for Metric-Based Cavity Remeshing and a Posteriori Curving of [formula omitted] Tetrahedral Meshes

A new high-order mesh correction method is introduced. Using the simplex algorithm, it maximizes directly the minimum of all control coefficients depending on a control point. This method provides the global optimum as quickly as to evaluate the min 8 times on average. The basic principle is applicable to all common element types (tetrahedra, hexahedra pyramids, prisms) and degrees. Meshes are corrected globally using this simplex-based Jacobian corrector in iterative smoothing.Riemannian edge length minimization is presented for metric-based high-order mesh curving. It is consistent with log-Euclidean metric interpolation, which is extended to high-order meshes. The interpolated metric field is differentiated analytically, with applications to other geometric quantities (quality, Jacobian conformity). Edge length minimization is fast and flexible as it can be used with metrics deriving from adaptation or e.g. propagated surface metrics.The cavity operator is a general topological operator that can apply insertions, collapses and generalized swaps. It is extended to P2 meshes through a modular approach with curvature applied in two steps. Prescribed curvature is the result of CAD/P3 surrogate projection on the boundary and of metric-based curvature in the interior. Necessary curvature results from simplex-based correction of the curved cavity. Both curvature schemes can easily be replaced by alternatives.Several 3D numerical results on difficult academic and realistic CFD test-cases are presented. P2 tetrahedral meshes with up to 20M tetrahedra are corrected with curved boundary in between 10 s and 3 m30 s depending on target minimum Jacobian determinant.

Combining High-Order Metric Interpolation and Geometry Implicitization for Curved r-Adaption

We detail how to use Newton's method for distortion-based curved $r$-adaption to a discrete high-order metric field while matching a target geometry. Specifically, we combine two terms: a distortion measuring the deviation from the target metric; and a penalty term measuring the deviation from the target boundary. For this combination, we consider four ingredients. First, to represent the metric field, we detail a log-Euclidean high-order metric interpolation on a curved (straight-edged) mesh. Second, for this metric interpolation, we detail the first and second derivatives in physical coordinates. Third, to represent the domain boundaries, we propose an implicit representation for 2D and 3D NURBS models. Fourth, for this implicit representation, we obtain the first and second derivatives. The derivatives of the metric interpolation and the implicit representation allow minimizing the objective function with Newton's method. For this second-order minimization, the resulting meshes simultaneously match the curved features of the target metric and boundary. Matching the metric and the geometry using second-order optimization is an unprecedented capability in curved (straight-edged) $r$-adaption. This capability will be critical in global and cavity-based curved (straight-edged) high-order mesh adaption.

High-Order Mixed Finite Element Variable Eddington Factor Methods

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the discrete ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory (LLNL). Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations.

High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries

We propose a method that morphs high-order meshes such that their boundaries and interfaces coincide/align with implicitly defined geometries. Our focus is particularly on the case when the target surface is prescribed as the zero isocontour of a smooth discrete function. Common examples of this scenario include using level set functions to represent material interfaces in multimaterial configurations, and evolving geometries in shape and topology optimization. The proposed method formulates the mesh optimization problem as a variational minimization of the sum of a chosen mesh-quality metric using the Target-Matrix Optimization Paradigm (TMOP) and a penalty term that weakly forces the selected faces of the mesh to align with the target surface. The distinct features of the method are use of a source mesh to represent the level set function with sufficient accuracy, and adaptive strategies for setting the penalization weight and selecting the faces of the mesh to be fit to the target isocontour of the level set field. We demonstrate that the proposed method is robust for generating boundary- and interface-fitted meshes for curvilinear domains using different element types in 2D and 3D.

Conservative high-order data transfer method on generalized polygonal meshes

A conservative data transfer (remap) between two meshes is an important step of arbitrary Lagrangian-Eulerian (ALE) hydrodynamics simulations. High-order numerical methods for ALE simulations require both high-order (curvilinear) meshes and high-order remap algorithms. We develop a conservative and bounds-preserving method for accurate remapping of discrete fields on generalized polygonal meshes with curvilinear edges. The properties of the proposed method are studied theoretically and numerically for various (smooth and non-smooth) mesh deformations and discrete fields that represent smooth and discontinuous functions.

Reducing errors caused by geometrical inaccuracy to solve partial differential equations with moving frames on curvilinear domain

In the numerical discretization of partial differential equations (PDEs) with moving frames on curved surfaces, the discretization error does not converge for a high p≥5. Moreover, the conservation error remains significant even in a refined mesh and does not converge as the polynomial order p increases. We postulate that the inaccurate location of the internal grid points of curved elements causes this problem; this is called the internal point error. This bottleneck of convergence persists even when the vertices of the elements are located exactly on the curved domain. We first theoretically explain the effects of the internal point error on numerical accuracy. We then computationally validate the postulation by using a high-order mesh with a negligible internal point error, even for a high-order polynomial of order p. For computational validations, four differential operators and four PDEs are numerically solved using moving frames on the unit sphere to demonstrate the effect of the internal point error.

Mesh‐free model for Hopf's bifurcation points in incompressible fluid flows problems

AbstractIn this article, a new numerical model is proposed to detect numerically the Hopf bifurcation point of incompressible fluid flows problems. The concept of this model consists to propose a Hopf bifurcation indicator which is solved by a high order mesh free algorithm (HO‐MFA) using the moving least squares (MLS) approximation. This numerical model is based on a strong formulation of Navier–Stokes equations. This procedure allowed us to recover the results of the literature in using only a modelization for low Reynolds number. This new model is tested on the classical flow around a cylindrical obstacle and the backward‐facing step flow to show the advantage and ability of the proposed model to detect the bifurcation points and to determinate the flow periodicity.

High-Order Cardiomyopathy Human Heart Model and Mesh Generation.

Faithful, accurate, and successful cardiac biomechanics and electrophysiological simulations require patient-specific geometric models of the heart. Since the cardiac geometry consists of highly-curved boundaries, the use of high-order meshes with curved elements would ensure that the various curves and features present in the cardiac geometry are well-captured and preserved in the corresponding mesh. Most other existing mesh generation techniques require computer-aided design files to represent the geometric boundary, which are often not available for biomedical applications. Unlike such methods, our technique takes a high-order surface mesh, generated from patient medical images, as input and generates a high-order volume mesh directly from the curved surface mesh. In this paper, we use our direct high-order curvilinear tetrahedral mesh generation method [1] to generate several second-order cardiac meshes. Our meshes include the left ventricle myocardia of a healthy heart and hearts with dilated and hypertrophic cardiomyopathy. We show that our high-order cardiac meshes do not contain inverted elements and are of sufficiently high quality for use in cardiac finite element simulations.

Bijective and coarse high-order tetrahedral meshes

We introduce a robust and automatic algorithm to convert linear triangle meshes with feature annotated into coarse tetrahedral meshes with curved elements. Our construction guarantees that the high-order meshes are free of element inversion or self-intersection. A user-specified maximal geometrical error from the input mesh controls the faithfulness of the curved approximation. The boundary of the output mesh is in bijective correspondence to the input, enabling attribute transfer between them, such as boundary conditions for simulations, making our curved mesh an ideal replacement or complement for the original input geometry. The availability of a bijective shell around the input surface is employed to ensure robust curving, prevent self-intersections, and compute a bijective map between the linear input and curved output surface. As necessary building blocks of our algorithm, we extend the bijective shell formulation to support features and propose a robust approach for boundary-preserving linear tetrahedral meshing. We demonstrate the robustness and effectiveness of our algorithm by generating high-order meshes for a large collection of complex 3D models.

Error-bounded Edge-based Remeshing of High-order Tetrahedral Meshes

We present a robust method to generate high-quality high-order tetrahedral meshes with bounded approximation errors and low mesh complexity. The success of our method relies on two key components. The first one is three novel local operations that robustly modify the topology of the high-order tetrahedral mesh while avoiding invalid (flipped or degenerate) elements. In practice, our meshing algorithm follows the edge-based remeshing algorithm that iteratively conducts these local topological operations and a geometric optimization operation to improve mesh quality. The second is a new containment check procedure that robustly judges whether the approximation error between the input mesh and the high-order mesh exceeds the user-specified bound. If one operation causes the error-bounded constraint to be violated, we reject this operation to ensure a bounded approximation error. Besides, the number of tetrahedrons of the high-order mesh is reduced by progressively increasing the target edge length in the edge-based remeshing algorithm. A large number of experimental results have shown the capability and feasibility of our method. Compared to other state-of-the-art methods, our method achieves higher robustness and quality.