Mesh Algorithm Research Articles

Diffuse interface immersed boundary method for multi-fluid flows with arbitrarily moving rigid bodies

We present an interpolation-free diffuse interface immersed boundary method for multiphase flows with moving bodies. A single fluid formalism using the volume-of-fluid approach is adopted to handle multiple immiscible fluids which are distinguished using the volume fractions, while the rigid bodies are tracked using an analogous volume-of-solid approach that solves for the solid fractions. The solution to the fluid flow equations are carried out using a finite volume-immersed boundary method, with the latter based on a diffuse interface philosophy. In the present work, we assume that the solids are filled with a “virtual” fluid with density and viscosity equal to the largest among all fluids in the domain. The solids are assumed to be rigid and their motion is solved using Newton's second law of motion. The immersed boundary methodology constructs a modified momentum equation that reduces to the Navier–Stokes equations in the fully fluid region and recovers the no-slip boundary condition inside the solids. An implicit incremental fractional-step methodology in conjunction with a novel hybrid staggered/non-staggered approach is employed, wherein a single equation for normal momentum at the cell faces is solved everywhere in the domain, independent of the number of spatial dimensions. The scalars are all solved for at the cell centres, with the transport equations for solid and fluid volume fractions solved using a high-resolution scheme. The pressure is determined everywhere in the domain (including inside the solids) using a variable coefficient Poisson equation. The solution to momentum, pressure, solid and fluid volume fraction equations everywhere in the domain circumvents the issue of pressure and velocity interpolation, which is a source of spurious oscillations in sharp interface immersed boundary methods. A well-balanced algorithm with consistent mass/momentum transport ensures robust simulations of high density ratio flows with strong body forces. The proposed diffuse interface immersed boundary method is shown to be discretely mass-preserving while being temporally second-order accurate and exhibits nominal second-order accuracy in space. We examine the efficacy of the proposed approach through extensive numerical experiments involving one or more fluids and solids, that include two-particle sedimentation in homogeneous and stratified environment. The results from the numerical simulations show that the proposed methodology results in reduced spurious force oscillations in case of moving bodies while accurately resolving complex flow phenomena in multiphase flows with moving solids. These studies demonstrate that the proposed diffuse interface immersed boundary method, which could be related to a class of penalisation approaches, is a robust and promising alternative to computationally expensive conformal moving mesh algorithms as well as the class of sharp interface immersed boundary methods for multibody problems in multi-phase flows.

Part‐Based Mesh Segmentation: A Survey

AbstractThis paper surveys mesh segmentation techniques and algorithms, with a focus on part‐based segmentation, that is, segmentation that divides a mesh (featuring a 3D object) into meaningful parts. Part‐based segmentation applies to a single object and also to a family of objects (i.e. co‐segmentation). However, we shall not address here chart‐based segmentation, though some mesh co‐segmentation methods employ such chart‐based segmentation in the initial step of their pipeline. Finally, the taxonomy proposed in this paper is new in the sense that one classifies each segmentation algorithm regarding the dimension (i.e. 1D, 2D and 3D) of the representation of object parts. The leading idea behind this survey is to identify the properties and limitations of the state‐of‐the‐art algorithms to shed light on the challenges for future work.

Design-order, non-conformal low-Mach fluid algorithms using a hybrid CVFEM/DG approach

A hybrid, design-order sliding mesh algorithm, which uses a control volume finite element method (CVFEM), in conjunction with a discontinuous Galerkin (DG) approach at non-conformal interfaces, is outlined in the context of a low-Mach fluid dynamics equation set. This novel hybrid DG approach is also demonstrated to be compatible with a classic edge-based vertex centered (EBVC) scheme. For the CVFEM, element polynomial, P, promotion is used to extend the low-order P=1 CVFEM method to higher-order, i.e., P=2. An equal-order low-Mach pressure-stabilized methodology, with emphasis on the non-conformal interface boundary condition, is presented. A fully implicit matrix solver approach that accounts for the full stencil connectivity across the non-conformal interface is employed. A complete suite of formal verification studies using the method of manufactured solutions (MMS) is performed to verify the order of accuracy of the underlying methodology. The chosen suite of analytical verification cases range from a simple steady diffusion system to a traveling viscous vortex across mixed-order non-conformal interfaces. Results from all verification studies demonstrate either second- or third-order spatial accuracy and, for transient solutions, second-order temporal accuracy. Significant accuracy gains in manufactured solution error norms are noted even with modest promotion of the underlying polynomial order. The paper also demonstrates the CVFEM/DG methodology on two production-like simulation cases that include an inner block subjected to solid rotation, i.e., each of the simulations include a sliding mesh, non-conformal interface. The first production case presented is a turbulent flow past a high-rate-of-rotation cube (Re, 4000; RPM, 3600) on like and mixed-order polynomial interfaces. The final simulation case is a full-scale Vestas V27 225 kW wind turbine (tower and nacelle omitted) in which a hybrid topology, low-order mesh is used. Both production simulations provide confidence in the underlying capability and demonstrate the viability of this hybrid method for deployment towards high-fidelity wind energy validation and analysis.

From 4D Medical Images (CT, MRI, and Ultrasound) to 4D Structured Mesh Models of the Left Ventricular Endocardium for Patient-Specific Simulations.

With cardiovascular disease (CVD) remaining the primary cause of death worldwide, early detection of CVDs becomes essential. The intracardiac flow is an important component of ventricular function, motion kinetics, wash-out of ventricular chambers, and ventricular energetics. Coupling between Computational Fluid Dynamics (CFD) simulations and medical images can play a fundamental role in terms of patient-specific diagnostic tools. From a technical perspective, CFD simulations with moving boundaries could easily lead to negative volumes errors and the sudden failure of the simulation. The generation of high-quality 4D meshes (3D in space + time) with 1-to-1 vertex becomes essential to perform a CFD simulation with moving boundaries. In this context, we developed a semiautomatic morphing tool able to create 4D high-quality structured meshes starting from a segmented 4D dataset. To prove the versatility and efficiency, the method was tested on three different 4D datasets (Ultrasound, MRI, and CT) by evaluating the quality and accuracy of the resulting 4D meshes. Furthermore, an estimation of some physiological quantities is accomplished for the 4D CT reconstruction. Future research will aim at extending the region of interest, further automation of the meshing algorithm, and generating structured hexahedral mesh models both for the blood and myocardial volume.

Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter

In this work, we consider an optimal accurate numerical approximation for the boundary layer adaptive solution of a system of reaction–diffusion problems involving a small perturbation parameter. To generate the layer adaptive mesh, we use the equidistribution of a positive monitor function. This technique enables the mesh movement to follow the rapidly changing behavior of the solution. It is observed that the discrete solution is second-order convergent on the layer adaptive equidistributed nonuniform mesh irrespective of the value of the perturbation parameter. This convergence rate is optimal with respect to the discretization of the continuous problem. We illustrate the performance of the present method based on a moving mesh algorithm by numerical experiments.

Robust structure simplification for hex re-meshing

We introduce a robust and automatic algorithm to simplify the structure and reduce the singularities of a hexahedral mesh. Our algorithm interleaves simplification operations to collapse sheets and chords of the base complex of the input mesh with a geometric optimization, which improves the elements quality. All our operations are guaranteed not to introduce elements with negative Jacobians, ensuring that our algorithm always produces valid hex-meshes, and not to increase the Hausdorff distance from the original shape more than a user-defined threshold, ensuring a faithful approximation of the input geometry. Our algorithm can improve meshes produced with any existing hexahedral meshing algorithm --- we demonstrate its effectiveness by processing a dataset of 194 hex-meshes created with octree-based, polycube-based, and field-aligned methods.

An efficient nonlinear octree SBFEM and its application to complicated geotechnical structures

An efficient nonlinear octree scaled boundary finite element method (SBFEM) is proposed to improve the ability of the SBFEM to analyse complicated structures. The polygon boundaries are directly solved using a pre-calculated mean-value interpolation. The nonlinearity can be considered via an internal quadratic tetrahedral integral scheme. Three simulations are performed to evaluate the accuracy, versatility, flexibility and high efficiency of the method. The difficulties of automatic modelling for both layered construction and material partitioning are eliminated using the octree meshing algorithm. The proposed method can serve as a powerful technique for the rapid design and analysis of geotechnical structures.

Simulation of the human airways using virtual topology tools and meshing optimization.

A method is proposed to improve the quality of the three-dimensional airway geometric models using a commercial software, checking the number of elements, meshing time, and aspect ratio and skewness parameters. The use of real and virtual topologies combined with patch-conforming and patch-independent meshing algorithms results in four different models being the best solution the combination of virtual topology and patch-independent algorithm, due to an excellent aspect ratio and skewness of the elements, and minimum meshing time. The result is a reduction in the computational time required for both meshing and simulation due to a smaller number of cells. The use of virtual topologies combined with patch-independent meshing algorithms could be extended in bioengineering because the geometries handling is similar to this case. The method is applied to a healthy person using their computed tomography images. The resulting numerical models are able to simulate correctly a forced spirometry.

Very high-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection–diffusion equation with Dirichlet condition

• Very high-order finite volume scheme for curved geometries. • Reconstruction for Off-site Data to provide the optimal order with curved domain. • No curved element and complex mapping on reference element. • A set numerical experiences with non-trivial geometries to assess the method order. Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the very high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers from several drawbacks, particularly, the use of curved elements that requires sophisticated meshing algorithms. We propose a new method where the physical domain and the computational domain are distinct and we introduce the Reconstruction for Off-site Data (ROD) where polynomial reconstructions are carried out on the mesh using data localized outside of the computational domain, namely the Dirichlet condition situated on the physical domain. A series of numerical tests assess the accuracy, convergence rates, robustness, and efficiency of the new method and show that the boundary condition is fully integrated in the scheme with a very high-order accuracy and the optimal convergence rate is achieved.

Monge–Ampére simulation of fourth order PDEs in two dimensions with application to elastic–electrostatic contact problems

We present an efficient moving mesh method for the simulation of fourth order nonlinear partial differential equations (PDEs) in two dimensions using the Parabolic Monge–Ampére (PMA) equation. PMA methods have been successfully applied to the simulation of second order problems, but not on systems with higher order equations which arise in many topical applications. Our main application is the resolution of fine scale behavior in PDEs describing elastic–electrostatic interactions. The PDE system considered has multiple parameter dependent singular solution modalities, including finite time singularities and sharp interface dynamics. We describe how to construct a dynamic mesh algorithm for such problems which incorporates known self similar or boundary layer scalings of the underlying equation to locate and dynamically resolve fine scale solution features in these singular regimes. We find a key step in using the PMA equation for mesh generation in fourth order problems is the adoption of a high order representation of the transformation from the computational to physical mesh. We demonstrate the efficacy of the new method on a variety of examples and establish several new results and conjectures on the nature of self-similar singularity formation in higher order PDEs.

Accurate halo–galaxy mocks from automatic bias estimation and particle mesh gravity solvers

Reliable extraction of cosmological information from clustering measurements of galaxy surveys requires estimation of the error covariance matrices of observables. The accuracy of covariance matrices is limited by our ability to generate sufficiently large number of independent mock catalogs that can describe the physics of galaxy clustering across a wide range of scales. Furthermore, galaxy mock catalogs are required to study systematics in galaxy surveys and to test analysis tools. In this investigation, we present a fast and accurate approach for generation of mock catalogs for the upcoming galaxy surveys. Our method relies on low-resolution approximate gravity solvers to simulate the large scale dark matter field, which we then populate with halos according to a flexible nonlinear and stochastic bias model. In particular, we extend the \textsc{patchy} code with an efficient particle mesh algorithm to simulate the dark matter field (the \textsc{FastPM} code), and with a robust MCMC method relying on the \textsc{emcee} code for constraining the parameters of the bias model. Using the halos in the BigMultiDark high-resolution $N$-body simulation as a reference catalog, we demonstrate that our technique can model the bivariate probability distribution function (counts-in-cells), power spectrum, and bispectrum of halos in the reference catalog. Specifically, we show that the new ingredients permit us to reach percentage accuracy in the power spectrum up to $k\sim 0.4\; \,h\,{\rm Mpc}^{-1}$ (within 5\% up to $k\sim 0.6\; \,h\,{\rm Mpc}^{-1}$) with accurate bispectra improving previous results based on Lagrangian perturbation theory.

A geometric mesh smoothing algorithm related to damped oscillations

We introduce a smoothing algorithm for triangle, quadrilateral, tetrahedral and hexahedral meshes whose centerpiece is a simple geometric triangle transformation. The first part focuses on the mathematical properties of the element transformation. In particular, the transformation gives rise directly to a continuous model given by a system of coupled damped oscillations. Derived from this physical model, adaptive parameters are introduced and their benefits presented. The second part discusses the mesh smoothing algorithm based on the element transformation and its numerical performance on example meshes.

Yet another hexahedral dominant meshing algorithm: HexDom

In this paper, we describe a robust meshing algorithm for obtaining a mixed mesh with large number of hexahedral/prismatic elements grown over the domain boundary respecting the user imposed anisotropic metric where physics matter the most and in areas where it is required to have the least number of elements. The inner section away from the boundaries is filled with the terminal octants of a non-conformal octree. The remaining unmeshed portion of the domain within the hexahedral/prismatic faces is filled with narrow bands of tetrahedra. The novel idea of the meshing algorithm is the formation of the cavity as slim as possible between the exposed faces of the outer most boundary layers and the octant faces of the inner most terminal octants, in such a way that the length scales of the cavity mesh spacings would allow the frontal tetrahedral meshing algorithm robustly succeeding to fill the cavity respecting its boundary faces without recovery issues. The algorithm could be applied to non-cubical, arbitrary geometries that can also be non-manifold. Each domain region is meshed recursively and within which, the tetrahedral filling algorithm constructs as many manifold cavity shells as the problem constrains are imposed by the boundary layers and the mesh size settings. The final hexahedral dominant mesh is exported to a face-based finite volume format (OpenFoam) so that the non-manifold nature of the mesh is captured by flux based numerical solvers consistently and accurately.

NFFT based Ewald summation for electrostatic systems with charges and dipoles

The efficient computation of Coulomb interactions in charged particle systems is of great importance in the field of molecular dynamics simulations. It is widely known that an approximation can be realized based on the Ewald summation approach and the fast Fourier transform (FFT). In the present paper we consider particle systems containing a mixture of N point charges as well as point dipoles. New cutoff errors in the Ewald summation formulas concerning charge–dipole interactions are derived and, moreover, validated by numerical examples. Furthermore, we present for the first time an O(Nlog⁡N) particle mesh algorithm for computing mixed charge–dipole interactions based on the FFT for nonequispaced data (NFFT). We present first numerical results for charge–dipole systems, showing that the introduced method can be tuned to a high precision and verifying the O(Nlog⁡N) scaling. In order to calculate the interactions with dipoles efficiently, two new variants of the NFFT, namely the Hessian NFFT as well as the adjoint gradient NFFT, are derived and implemented. In the context of NFFT, these new variants are of great importance on their own. The presented particle mesh method is an extension of the particle–particle NFFT (P2NFFT) framework. Therefore, all the formerly derived P2NFFT features, which cover for instance the treatment of arbitrary combinations of periodic and non-periodic boundary conditions, the handling of triclinic box shapes and a massively parallel implementation, are now also supported for mixed charge–dipole as well as pure dipole systems. The algorithms are publicly available as a part of the ScaFaCoS library.

Ternary Sparse Matrix Representation for Volumetric Mesh Subdivision and Processing on GPUs

AbstractIn this paper, we present a novel volumetric mesh representation suited for parallel computing on modern GPU architectures. The data structure is based on a compact, ternary sparse matrix storage of boundary operators. Boundary operators correspond to the first‐order top‐down relations of k‐faces to their (k − 1)‐face facets. The compact, ternary matrix storage format is based on compressed sparse row matrices with signed indices and allows for efficient parallel computation of indirect and bottom‐up relations. This representation is then used in the implementation of several parallel volumetric mesh algorithms including Laplacian smoothing and volumetric Catmull‐Clark subdivision. We compare these algorithms with their counterparts based on OpenVolumeMesh and achieve speedups from 3× to 531×, for sufficiently large meshes, while reducing memory consumption by up to 36%.

Robust hex-dominant mesh generation using field-guided polyhedral agglomeration

We propose a robust and efficient field-aligned volumetric meshing algorithm that produces hex-dominant meshes, i.e. meshes that are predominantly composed of hexahedral elements while containing a small number of irregular polyhedra. The latter are placed according to the singularities of two optimized guiding fields, which allow our method to generate meshes with an exceptionally high amount of isotropy. The field design phase of our method relies on a compact quaternionic representation of volumetric octa-fields and a corresponding optimization that explicitly models the discrete matchings between neighboring elements. This optimization naturally supports alignment constraints and scales to very large datasets. We also propose a novel extraction technique that uses field-guided mesh simplification to convert the optimized fields into a hexdominant output mesh. Each simplification operation maintains topological validity as an invariant, ensuring manifold output. These steps easily generalize to other dimensions or representations, and we show how they can be an asset in existing 2D surface meshing techniques. Our method can automatically and robustly convert any tetrahedral mesh into an isotropic hex-dominant mesh and (with minor modifications) can also convert any triangle mesh into a corresponding isotropic quad-dominant mesh, preserving its genus, number of holes, and manifoldness. We demonstrate the benefits of our algorithm on a large collection of shapes provided in the supplemental material along with all generated results.

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

High order compact scheme has extensively been used for solving the Helmholtz equation on uniform mesh. Although uniform spacing is computationally advantageous for isotropic problems, the non-uniform mesh will be superior to the problems with the anisotropic properties or boundary layers. In this paper, we proposed a compact scheme on nonuniform mesh which has at least third order accuracy for the 2D Helmholtz equation with variable wave number. A novel adaptive mesh algorithm is also presented to generate the orthogonal nonuniform mesh. The 2D Helmholtz equation is solved by combining the HOC scheme on nonuniform mesh with the moving mesh method. The multigrid method is used to solve the discrete algebraic system. Numerical experiments are executed to verify the accuracy and efficiency of the presented method.

The Relationship Between Surface Curvature and Abdominal Aortic Aneurysm Wall Stress.

The maximum diameter (MD) criterion is the most important factor when predicting risk of rupture of abdominal aortic aneurysms (AAAs). An elevated wall stress has also been linked to a high risk of aneurysm rupture, yet is an uncommon clinical practice to compute AAA wall stress. The purpose of this study is to assess whether other characteristics of the AAA geometry are statistically correlated with wall stress. Using in-house segmentation and meshing algorithms, 30 patient-specific AAA models were generated for finite element analysis (FEA). These models were subsequently used to estimate wall stress and maximum diameter and to evaluate the spatial distributions of wall thickness, cross-sectional diameter, mean curvature, and Gaussian curvature. Data analysis consisted of statistical correlations of the aforementioned geometry metrics with wall stress for the 30 AAA inner and outer wall surfaces. In addition, a linear regression analysis was performed with all the AAA wall surfaces to quantify the relationship of the geometric indices with wall stress. These analyses indicated that while all the geometry metrics have statistically significant correlations with wall stress, the local mean curvature (LMC) exhibits the highest average Pearson's correlation coefficient for both inner and outer wall surfaces. The linear regression analysis revealed coefficients of determination for the outer and inner wall surfaces of 0.712 and 0.516, respectively, with LMC having the largest effect on the linear regression equation with wall stress. This work underscores the importance of evaluating AAA mean wall curvature as a potential surrogate for wall stress.

Improved HPC method for nonlinear wave tank

The recently developed Harmonic Polynomial Cell (HPC) method has been proved to be a promising choice for solving potential-flow Boundary Value Problem (BVP). In this paper, a flux method is proposed to consistently deal with the Neumann boundary condition of the original HPC method and enhance the accuracy. Moreover, fixed mesh algorithm with free surface immersed is developed to improve the computational efficiency. Finally, a two dimensional (2D) multi-block strategy coupling boundary-fitted mesh and fixed mesh is proposed. It limits the computational costs and preserves the accuracy. A fully nonlinear 2D numerical wave tank is developed using the improved HPC method as a verification.

Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake

Tip vortex cavitation is the first type of cavitation to take place around most marine propellers. But the numerical prediction of tip vortex cavitation is one of the challenges for propeller wake because of turbulence dissipation during the numerical simulation. Several parameters of computational mesh and numerical algorithm are tested by mean of the predicted length of tip vortex cavtiation to validate a developed method. The predicted length of tip vortex cavtiation is on the increase about 0.4 propeller diameters using the developed numerical method. The predicted length of tip vortex cavtiation by RNG k – e model is about 3 times of that by SST k – ω model. Therefore, based on the validation of the present approach, the cavitating flows generated by two rotating propellers under a non-uniform inflow are calculated further. The distributions of axial velocity, total pressure and vapor volume fraction in the transversal planes across tip vortex region are shown to be useful in analyzing the feature of the cavitating flow. The strongest kernel of tip vortex cavitation is not at the position most close to blade tip but slightly far away from the region. During the growth of tip vortex cavitation extension, it appears short and thick, and then it becomes long and thin. The pressure fluctuations at the positions inside tip vortex region also validates the conclusion. A key finding of the study is that the grids constructed especially for tip vortex flows by using separated computational domain is capable of decreasing the turbulence dissipation and correctly capturing the feature of propeller tip vortex cavitation under uniform and non-uniform inflows. The turbulence model and advanced grids is important to predict tip vortex cavitation.