Function Of Mesh Size Research Articles

An automatic mesh generator for coupled 1D–2D hydrodynamic models

Abstract. Two-dimensional (2D), depth-averaged shallow water equation (SWE) models are routinely used to simulate flooding in coastal areas – areas that often include vast networks of channels and flood-control topographic features and/or structures, such as barrier islands and levees. Adequately resolving these features within the confines of a 2D model can be computationally expensive, which has led to coupling 2D simulation tools to less expensive one-dimensional (1D) models. Under certain 1D–2D coupling approaches, this introduces internal constraints that must be considered in the generation of the 2D computational mesh used. In this paper, we further develop an existing automatic unstructured mesh generation tool for SWE models, ADMESH+, to sequentially (i) identify 1D constraints from the raw input data used in the mesh generation process, namely the digital elevation model (DEM) and land–water delineation data; (ii) distribute grid points along these internal constraints, according to feature curvature and user-prescribed minimum grid spacing; and (iii) integrate these internal constraints into the 2D mesh size function and mesh generation processes. The developed techniques, which include a novel approach for determining the so-called medial axis of a polygon, are described in detail and demonstrated on three test cases, including two inland watersheds with vast networks of channels and a complex estuarine system on the Texas, USA, coast.

Accelerating turbulent reacting flow simulations on many-core/GPUs using matrix-based kinetics

The present work assesses the impact - in terms of time to solution, throughput analysis, and hardware scalability - of transferring computationally intensive tasks, found in compressible reacting flow solvers, to the GPU. Attention is focused on outlining the workflow and data transfer penalties associated with “plugging in” a recently developed GPU-based chemistry library into (a) a purely CPU-based solver and (b) a GPU-based solver, where, except for the chemistry, all other variables are computed on the GPU. This comparison allows quantification of host-to-device (and vice versa) data transfer penalties on the overall solver speedup as a function of mesh and reaction mechanism size. To this end, a recently developed GPU-based chemistry library known as UMChemGPU is employed to treat the kinetics in the flow solver KARFS. UMChemGPU replaces conventional CPU-based Cantera routines using a matrix-based formulation. The impact of i) data transfer times, ii) chemistry acceleration, and iii) the hardware architecture is studied in detail in the context of GPU saturation limits. Hydrogen and dimethyl ether (DME) reaction mechanisms are used to assess the impact of the number of species/reactions on overall/chemistry-only speedup. It was found that offloading the source term computation to UMChemGPU results in up to 7X reduction in overall time to solution and four orders of magnitude faster source term computation compared to conventional CPU-based methods. Furthermore, the metrics for achieving maximum performance gain using GPU chemistry with an MPI + CUDA solver are explained using the Roofline model. Integrating the UMChemGPU with an MPI + OpenMP solver does not improve the overall performance due to the associated data copy time between the device (GPU) and host (CPU) memory spaces. The performance portability was demonstrated using three different GPU architectures, and the findings are expected to translate to a wide variety of high-performance codes in the combustion community.

Inverting the discrete curl operator: A novel graph algorithm to find a vector potential of a given vector field

We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse theory. We introduce the new concept of complete acyclic matchings and we show that they give the same end result of Gaussian elimination. Basically, instead of doing costly row and column operations on a sparse matrix, we compute equivalent cheap combinatorial operations that preserve the underlying sparsity structure. Currently, the most efficient algorithms proposed in literature to find discrete vector potentials make use of tree-cotree techniques. We show that they compute a special type of complete acyclic matchings. Moreover, we show that the problem of computing them is equivalent to the problem of deciding whether a given mesh has a topological property called collapsibility. This fact gives a topological characterization of well-known termination problems of tree-cotree techniques. We propose a new recursive algorithm to compute discrete vector potentials. It works directly on basis elements of 1- and 2-chains by performing elementary Gaussian operations on them associated with acyclic matchings. However, the main novelty is that it can be applied recursively. Indeed, the recursion process allows us to sidetrack termination problems of the standard tree-cotree techniques. We tested the algorithm on pathological triangulations with known topological obstructions. In all tested problems we observe linear computational complexity as a function of mesh size. Moreover, the algorithm is purely graph-based so it is straightforward to implement and does not require specialized external procedures. We believe that our framework could offer new perspectives to sparse matrix computations.

Direct exchange calculation for unstructured micromagnetic meshes

In micromagnetic simulations it is necessary to compute the effective field associated to the exchange interaction, which corresponds to applying the Laplace operator to the magnetization field. The most widely used approach for computing the exchange field is to apply finite-difference schemes to homogeneous Cartesian meshes. In many situations it would be computationally advantageous to perform micromagnetic simulation on inhomogeneous meshes, but doing so presents a challenge with regard to the calculation of the exchange field. Here we present the construction of an exchange field calculation method for inhomogeneous meshes. A novel second order differential operator method is introduced and applied to a series of analytical functions on irregular meshes and compared to existing methods, finding an accuracy improvement of up to 81% relative to the second order finite difference method. The method is then used in combination with the MagTense framework and tested on the μmag standard micromagnetics problems 3 and 4. Here we find that especially for tetrahedral meshes, the proposed method results in much faster convergence as function of mesh size. We demonstrate that the method converges stably even for meshes where some mesh elements have a volume as small as 1/8 compared to the base resolution. Finally, we show that the method works equally well for meshes with spatially varying material parameters.

Nonlinear time‐domain wave‐structure interaction: A parallel fast integral equation approach

AbstractWe report on the development and validation of a new numerical wave tank (NWT) solving fully nonlinear potential flow (FNPF) equations, as a more efficient variation of Grilli et al.'s NWT [Grilli et al., A fully nonlinear model for three‐dimensional overturning waves over arbitrary bottom. Int J Numer Methods Fluids. 2001; 35:829‐867], which was successful at modeling many wave phenomena, including landslide‐generated tsunamis, rogue waves, and the initiation of wave breaking over slopes. This earlier NWT combined a three dimensional MII (mid‐interval interpolation) boundary element method (BEM) to an explicit mixed Eulerian–Lagrangian time integration. The latter was based on second‐order Taylor series expansions for the mesh geometry and Dirichlet free surface boundary condition for the potential, requiring high‐order derivatives to be computed in space and time. Here, to be able to solve large scale wave‐structure interaction problems for surface‐piercing bodies of complex geometry, of interest for ocean engineering and naval hydrodynamics applications, the NWT is reformulated to use cubic B‐spline meshes and the BEM solution is accelerated with a parallelized fast multipole method (FMM) based on ExaFMM, one of the fastest open source FMM to date. The NWT accuracy, convergence, and scaling are first assessed for simple cases, by comparing results with those of the earlier MII‐NWT as a function of mesh size and other model parameters. The relevance of the new NWT for solving the targeted applications is then demonstrated for surface piercing fixed cylinders, for which we show that results agree well with theoretical and experimental data for wave elevation and hydrodynamic forces.

Improving convergence by optimizing the condition number of the stiffness matrices arising from least-squares finite element methods

The least-squares finite element method (LSFEM) has drawn much attention with desirable properties, such as always symmetric positive-definite stiffness matrix and approximation of primal and non-primal variables with equivalent accuracy. Despite a condition number comparable to that of Galerkin FEM, of O(h−2), it is sometimes found that LSFEM with low-order elements does not show a satisfactory accuracy of solution. In this report, we propose an easy-to-implement technique that optimizes the condition number of the stiffness matrix. The least-squares functional is applied to the first-order governing equations and a linear combination of the weak form domain and boundary equations, with normalization (i.e., weighting) parameters is constructed. The impact of these parameters on the condition number of the resulting global stiffness matrix is proved to be independent of mesh size or shape function order. By use of a global optimization scheme that tunes these parameters, we observe notable decrease in the condition number compared to cases where the least-squares principle is directly applied to the governing equations, and hence convergence is significantly improved. It is further shown that the technique can be applied to the elemental stiffness matrix for a computationally efficient means of determining normalization parameters. To show the general applicability of the method, we also apply this technique to solve for a two-dimensional (2D) lid-driven flow problem and to calculate the electromagnetic field distribution in a 2D rectangular resonant cavity. In both problems, the LSFEM solution accuracy is poor prior to optimization of weighting parameters, and accuracy does not increase much by mesh refinement alone. Moreover, the process can be generalized to Galerkin FEM and other methods to improve convergence properties.

Smooth Gradation of Anisotropic Meshes Using Log–Euclidean Metrics

The anisotropic mesh-size function represented by metric tensors includes two features: mesh sizes and mesh orientation. Such metrics are widely used in scientific computing for adaptation, but the solution metrics are not smooth, which adversely affects adaptation. A novel algorithm is proposed in this paper to smooth the metric as a whole and to improve anisotropic mesh-size gradation. First, the concept of Log–Euclidean metrics is used to convert the metric tensors from the Riemannian space into a Euclidean space. Then, the variations of metric tensors are limited by constraining the gradients of metric tensors in this space over the region of each background mesh element. Finally, a convex nonlinear optimization problem is formulated to smooth the metric tensors over the entire meshing domain. Theoretical analysis reveals the existence of a globally optimal smoothed sizing function. Numerical experiments on anisotropic meshes in both analytical and real laminar and turbulent flow cases are presented to demonstrate the effectiveness of the proposed method.

A Novel Meshing Method Based on Adaptive Size Function and Moving Mesh for Electromagnetic Finite Element Analysis

A moving meshing algorithm with mesh adaptive size function was proposed in this paper with regard to the modeling speed and solution accuracy of electromagnetic equipment in the optimization design process. In the proposed method, a mesh size function that considers curvature, feature size, and distance gradient restrictions is constructed, which can obtain high quality meshes and avoid excessive iteration; when the finite element mesh domain is deformed, only the mesh nodes close to the moving boundary are allowed to move, and the theory of force-balance is used combined with the second-order boundary projection algorithm to perform iterative optimization of the mesh node positions. The proposed method has the advantages of keeping the original mesh structure and minimum mesh deformation as well as speed up the convergence, save time in the finite element meshing, and ensure the quality of the generated mesh. Then, the proposed method was applied to a 37 kw motor for electromagnetic analysis, and the results obtained proved the accuracy of the algorithm; finally, the effectiveness of the mesh movement algorithm in three-dimensional space was tested by moving the sphere inside the cylinder.

Hydrogel Properties May Influence Mesenchymal Stem Cell Lineage Progression Through Modulating GAPDH Activity

Cell traction forces, biochemical signals, and cell metabolism are each known to regulate mesenchymal stem cell (MSC) differentiation. Biomaterials have therefore been designed to manipulate cell traction force (via their viscoelasticity and adhesion ligands) to guide MSC fate decisions. Similarly, the type and density of biochemical signals presented by a material have been tailored to influence MSC differentiation. However, the potential impact of biomaterial properties on regulating MSC differentiation through modulating cell metabolism is relatively unstudied. Here, we present data indicating that hydrogel elastic modulus and mesh size regulate MSC differentiation in part through modulating the activity of the metabolic enzyme glyceraldehyde-3-phosphate-dehydrogenase (GAPDH). Toward this end, we first confirm that the differentiation profile of MSCs cultured in 2D on highly elastic, covalently crosslinked poly(ethylene glycol) diacrylate (PEGDA) hydrogels differs substantially from that of MSCs encapsulated within the same hydrogel formulations, indicating a dependence in 3D on a variable(s) beyond elastic modulus. Further results indicate that the GAPDH activity of MSCs in 3D hydrogels is a function of elastic modulus and mesh size, suggesting that GAPDH activity may be one of these variables. Studies in 2D supported a positive correlation between hydrogel elastic modulus and GAPDH activity. Additionally, inhibition of GAPDH activity on 2D surfaces induced alterations in the profiles of key differentiation markers, indicating that GAPDH activity can impact lineage progression. Cumulatively, these findings suggest that the potential impact of hydrogel properties on cell metabolism should be considered when evaluating biomaterial-driven MSC differentiation. Cell traction forces, biochemical signals, and cell metabolism are each known to regulate mesenchymal stem cell (MSC) differentiation. Biomaterials have therefore been designed to manipulate cell traction force (via their viscoelasticity and adhesion ligands) to guide MSC fate decisions. Similarly, the type and density of biochemical signals presented by a material have been tailored to influence MSC differentiation. However, the impact of biomaterial properties on regulating MSC differentiation through modulating cell metabolism is relatively unstudied. Here, we present data indicating that hydrogel elastic modulus and mesh size regulate MSC differentiation in part through modulating the activity of the metabolic enzyme glyceraldehyde-3-phosphate-dehydrogenase (GAPDH).

Algorithm for Curved Surface Mesh Generation Based on Delaunay Refinement

Curved surface mesh generation is a key step for many areas. Here, a mesh generation algorithm for closed curved surface based on Delaunay refinement is proposed. We focus on improving the shape quality of the meshes generated and making them conform to 2-manifold. The Delaunay tetrahedralization of initial sample is generated first, the initial surface mesh which is a subset of the Delaunay tetrahedralization can be achieved. A triangle is refined by inserting a new point if it is large or of bad quality. For each sample, we also check the triangles that adjoin it whether from a topological disk. If not, the largest triangle will be refined. Finally, the surface mesh is updated after a new point is inserted into the sample. The definition of mesh size function for surface mesh generation is also put in this paper. Meshing experiments of some models demonstrate that the new algorithm is advantageous in generating high quality surface mesh, the count of mesh is suitable and can well approximate the curved surface. The presented method can be used for a wide range of problems including computer graphics, computer vision and finite element method.

OceanMesh2D 1.0: MATLAB-based software for two-dimensional unstructured mesh generation in coastal ocean modeling

Abstract. OceanMesh2D is a set of MATLAB functions with preprocessing and post-processing utilities to generate two-dimensional (2-D) unstructured meshes for coastal ocean circulation models. Mesh resolution is controlled according to a variety of feature-driven geometric and topo-bathymetric functions. Mesh generation is achieved through a force balance algorithm to locate vertices and a number of topological improvement strategies aimed at improving the worst-case triangle quality. The placement of vertices along the mesh boundary is adapted automatically according to the mesh size function, eliminating the need for contour simplification algorithms. The software expresses the mesh design and generation process via an objected-oriented framework that facilitates efficient workflows that are flexible and automatic. This paper illustrates the various capabilities of the software and demonstrates its utility in realistic applications by producing high-quality, multiscale, unstructured meshes.

NUMERICAL ANALYSIS OF EFFECTIVENESS FOR VEHICLE NET SYSTEMS PROTECTING AGAINST SHAPED CHARGE PROJECTILES

Analysis of effectiveness of two types of net systems protecting vehicles against projectiles with cumulative (shaped charge) warhead is presented in the article. Numerical models of net screens and PG7-WM projectile were created to simulate projectile penetration into the screen. Numerical analyses were used to determine a maximal distance in radial direction from projectile axis for which screen elements should come into contact with the projectile to neutralize it. Determined distances were used to calculate the probability of neutralization of pro-jectiles by net systems. Calculations were performed for elevation and azimuth angles of α = 0÷45°. The values calculated in such way were used to make graphs of PG-7WM projectile neutralization probability for both types of net systems in function of mesh size and azimuth and elevation angles at various mesh sizes. Received results may be used to optimize geometries of net systems to in-crease effectiveness of projectile neutralization.

Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141–153, 2016), we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms.

A new approach to automatic and a priori mesh adaptation around circular holes for finite element analysis

Through our research on the integration of finite element analysis in the design and manufacturing process with CAD, we have proposed the concept of mesh pre-optimization. This concept consists in converting shape and analysis information in a size map (a mesh sizing function) with respect to various adaptation criteria (refining the mesh around geometric form features, minimizing the geometric discretization error, boundary conditions, etc.). This size map then represents a constraint that has to be respected by automatic mesh generation procedures. This paper introduces a new approach to automatic mesh adaptation around circular holes. This tool aims at optimizing, before any FEA, the mesh of a CAD model around circular holes. This approach, referred to as “a priori” mesh adaptation, should not be regarded as an alternative to adaptive a posteriori mesh refinement but as an efficient way to obtain reasonably accurate FEA results before a posteriori adaptation, which is particularly interesting when evaluating design scenarios. The approach is based on performing many offline FEA analyses on a reference case and deriving, from results and error distributions obtained, a relationship between mesh size and FEA error. This relationship can then be extended to target user specified FEA accuracy objectives in a priori mesh adaptation for any distribution of circular holes. The approach being purely heuristic, fulfilling FEA accuracy objectives, in all cases, cannot be theoretically guaranteed. However, results obtained using varying hole diameters and distributions in 2D show that this heuristic approach is reliable and useful. Preliminary results also show that extension of the method can be foreseen towards a priori mesh adaptation in 3D and mesh adaptation around other types of 2D features.

Experimental Study and Mathematical Modeling of Propane-SCR-NOx Using Group Method of Data Handling and Artificial Neural Network

Abstract In this study, the catalytic behavior of protonated clinoptilolite in propane-SCR-NOx was investigated. The experiments were carried out in the temperature range of 200–500 °C as a function of zeolite mesh size 20, 35 and 70 at different weights of zeolite (0.45–1 g) and flow rates (300–600 ml/min) and consequently at various gas hourly space velocities (GHSV). Group method of data handling (GMDH) and artificial neural network (ANN) system were applied for mathematical modeling of NOx conversion to N2 in propane-SCR-NOx. The operating temperature (T), volumetric flow rate (F) and the weight of clinoptilolite zeolite (W) and the conversion of NOx to N2 (X) were considered as the inputs and output, respectively. In order to evaluate the models performance, conversions of NOx obtained from the GMDH and ANN systems were compared with those obtained from the experimental method. It is concluded that the ANN could successively estimate the conversion and the results were in a good agreement with the experimental data.

Optimal grading of the newest vertex bisection andH1-stability of theL2-projection

We show for adaptive triangulations in two dimensions, which are generated by the newest vertex bisection, an optimal grading estimate. Roughly speaking, we construct from the piecewise constant mesh-size function a regularized one with the following two properties. First, the two functions are equivalent, and second, the regularized mesh-size function differs at most by a factor of 2 on neighbouring elements. In combination with Bank & Yserentant (2014, Numer. Math.126, 361–381), this optimal grading estimate enables us to show that the |$L_2$|-orthogonal projections onto the space of continuous Lagrange finite elements up to order 12 is |$H^1$|-stable. We extend these results to a modified red–green refinement.

A coarse-grained molecular dynamics study of segmental structure and mobility in capped crosslinked copolymer films.

We present results from molecular-dynamics simulations of a generic bead-spring model of copolymer chains confined between solid walls and report on the glass-transition temperature and segmental dynamics as a function of film thickness and mesh size (the end-to-end distance of the subchains in the crosslinked polymer networks). Apparently, the glass-transition temperature displayed a steep increase for mesh-size values much smaller than the radius of gyration of the bulk chains, otherwise it remained invariant to mesh-size variations. The rise in the glass-transition temperature with decreasing mesh size and film thickness was accompanied by a monotonic slowing-down of segmental dynamics on all studied length scales. This observation is attributed to the correspondingly decreased width of the bulk density layer that was obtained in films whose thickness was larger than the end-to-end distance of the bulk polymer chains. To test this hypothesis, additional simulations were performed in which the crystalline walls were replaced with amorphous or rough walls. In the amorphous case, the high polymer density close to the walls vanished, but the dynamic response of the film was not affected. The rough walls, on the other hand, only slightly decreased the density close to the walls and led to a minor slowing-down in the dynamics at large length-scales.

Second-order electromagnetic eigenfrequencies of a triaxial ellipsoid II

Finite-element calculations of electromagnetic eigenvalues are known to converge to the exact solutions in the limit of vanishing element sizes. In an extension of previous work (Mehl 2009 Metrologia 46 554–9) the eigenfrequencies of the TM1n and TE1n (n = 1, 2, ··· 6) modes of triaxial ellipsoids were calculated as a function of mesh size. Higher-accuracy eigenvalues were obtained through a limiting process as the mesh size was reduced; the extrapolation was based on the theoretical convergence rate. The difference between the finite-element eigenfrequencies and the eigenfrequencies predicted by shape perturbation theory is found to be proportional to the cube of the fractional deformation parameter ϵ for all investigated modes. For ellipsoids with axes proportional to 1 : 1.0005 : 1.0010, the cubic term represents a fractional perturbation of the average TM16 eigenvalue k2 by −0.16 × 10 − 6 and the average TE16 eigenvalue by −0.22 × 10 − 6. This work adds support to the correctness of the analytic second-order formula derived in the previous work, and also demonstrates the usefulness of finite-element methods for investigating the quasi-spherical resonators (QSRs) used in measurements of the Boltzmann constant. In principle, the method can be extended to QSRs whose shape differs from triaxial ellipsoids.

TH-AB-BRA-10: Clinical Implementation of a Grid-Based Boltzmann Solver with Adaptive Meshing for Nuclear Medicine Dosimetry

Purpose: We previously compared the Grid-Based Boltzmann Solver (GBBS) with DOSXYZnrc Monte Carlo (MC) for beta and gamma sources in uniform materials and along material interfaces. The current standard in nuclear medicine is MIRD, which only calculates mean organ doses assuming uniform uptake in anthropomorphic phantoms. This work extends GBBS to clinical nuclear medicine for fast and accurate patient-specific voxel-level dosimetry. Methods: We compared the GBBS with MC using a patient’s post-therapy quantitative 90Y microsphere bremsstrahlung SPECT/CT.The GBBS (Attila™ v8.0.0) requires a tetrahedral mesh as input instead of voxels. Adaptive tetrahedral meshes were generated with TetGen v1.5.0 by folding CT and SPECT images into a mesh sizing function; the function was defined at each node on a background mesh with nodes defined at each SPECT/CT voxel. Target tetrahedral edge lengths varied from 0.5 to 8 cm and were adapted on SPECT activity level, CT material, and material gradient.GBBS used 30 electron energy groups, 32 angles, and 67,000 tetrahedrons. GBBS reported point data aligned with the center of SPECT/CT voxels which match MC. We report: 1) run time of the GBBS, 2) average difference in non-zero absorbed dose voxels, and 3) percent difference in mean absorbed dose to tumor and normal liver. DVHs and isodose curves were visually compared, and the tetrahedral mesh was compared to SPECT/CT. Results: GBBS calculation required 2 minutes. The average difference over all non-zero absorbed dose voxels (N=318,000) was −0.2 Gy. GBBS mean doses agreed within 1% of MC for both normal liver (39.1 Gy vs 39.3 Gy) and tumor (317 Gy vs 320 Gy). Adaptive meshing allowed sufficient material assignment and yielded smaller tetrahedrons near high activity regions and bone. DVHs had excellent visual agreement. Conclusions: GBBS with adaptive meshing is practical for fast and accurate clinical nuclear medicine dosimetry. Research reported in this publication was supported by the National Cancer Institute of the National Institutes of Health under Award Number R01CA138986. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

A Matlab mesh generator for the two-dimensional finite element method

A new Matlab code for the generation of unstructured (3-node or 6-node) triangular meshes in two dimensions is proposed. The method is based on the Matlab mesh generator distmesh of Persson and Strang (2004). As input, the code takes a signed distance function for the domain geometry. A mesh size function, for the spatial node distribution, is constructed using an approximate medial axis. As outputs, the code generates a 3-node or a 6-node triangular mesh with boundary data (edges and nodes). The approach presented consists of three steps: (1) an initial nodes placement is obtained using a probabilistic node distribution, (2) an iterative smoothing is performed assuming the presence of an attractive/repulsive internode force, and (3) a fast refinement procedure is performed for 6-node triangular meshes or large scale meshes.