Irregular Meshes Research Articles

Improvement of the Results of Finite Element Method in Plate Analysis Using Mesh Sizing Modifying Function

In the finite element methods (FEM), the mesh dimension, and the number of elements can affect the responses of structures. In this paper, a procedure is proposed to modify the stiffness matric of the plate element based on the mesh geometry and mesh size for reducing the central deflection error. For this purpose, the modifying coefficients are defined for the bending and shear stiffness matrices. The sensitivity of coefficients is investigated when the thickness of the plate element, the mesh dimension, and also the type of supports vary. The analysis results prove that the values of the bending coefficient have more effect in comparison to the shear coefficient. Finally, a function is proposed to determine the bending and shear matric coefficients based on the values of the exact displacements. Various numerical studies indicate that the modifying function has significantly improved the performance of the plate element, especially for the plates with irregular and large mesh dimensions.

New hybrid compact schemes for structured irregular meshes using Birkhoff polynomial basis

This paper discusses the formulation of a class of hybrid compact schemes for structured irregular meshes. First, we derived three- and five-point compact schemes with required boundary closures for non-periodic problems. Second, we devised hybrid compact schemes from the explicit representation of the first- and second-order derivatives on irregular staggered and collocated meshes, respectively. For staggered meshes, hybrid interpolation schemes are also formulated. The main advantage of the hybrid formulation is that it requires boundary closures only for the compact scheme for first-order derivatives on the collocated mesh. Resolution and added numerical diffusion properties of hybrid schemes are assessed by using global spectral analysis (Sengupta et al., 2003). Direct numerical simulations of 2D dispersive shallow water and incompressible Navier-Stokes equations at different Reynolds numbers with fourth-order RK4 time integration method are considered to validate the accuracy and efficiency of new schemes. Numerical solutions are also compared with results reported in the literature.

Phong deformation

Physics-based simulations of deforming tetrahedral meshes are widely used to animate detailed embedded geometry. Unfortunately most practitioners still use linear interpolation (or other low-order schemes) on tetrahedra, which can produce undesirable visual artifacts, e.g., faceting and shading artifacts, that necessitate increasing the simulation's spatial resolution and, unfortunately, cost. In this paper, we propose Phong Deformation , a simple, robust and practical vertex-based quadratic interpolation scheme that, while still only C 0 continuous like linear interpolation, greatly reduces visual artifacts for embedded geometry. The method first averages element-based linear deformation models to vertices, then barycentrically interpolates the vertex models while also averaging with the traditional linear interpolation model. The method is a fast, robust, and easily implemented replacement for linear interpolation that produces visually better results for embedded deformation with irregular tetrahedral meshes.

Cracking elements method with 6-node triangular element

The cracking elements method (CEM) is a novel Galerkin-based numerical approach for simulating cracking and fracturing processes. It is a crack-opening approach that avoids precise descriptions of the mechanical states of crack tips and captures the initiations and propagations of multiple cracks without nodal enrichment or crack tracking. The CEM requires element types with nonlinear interpolation of the displacement field to avoid stress-locking. In the 2D condition, the 6-node triangular element (T6) and 8-node quadrilateral element (Q8) are potential candidates. However, despite the success of the formerly proposed CEM with Q8, the CEM with T6 showed considerable mesh dependencies. In this work, to solve this problem, the CEM with T6 is further investigated. The mesh dependencies are shown to be eliminated with simple modification to the real characteristic length of the T6 element in the CEM framework. Several numerical examples with regular and irregular Q8 and T6 mixed meshes are provided, indicating the effectiveness and robustness of this approach.

Novel implementation of extrusion constraint in topology optimization by Helmholtz-type anisotropic filter

In this paper, a novel implementation of extrusion constraint in topology optimization is proposed based on Helmholtz-type anisotropic filter approach. The main idea is to set a far larger filter feature size along the extrusion direction than other directions, and thus, the density field is kept constant along the extrusion direction. The Helmholtz-type anisotropic filter can be easily implemented by adding a few more parameters to the isotropic one. Three illustrative examples, including 3D cantilever, spherical frame, and S-shape curved shell, are carried out to verify the effectiveness and robustness of the proposed method. Example results corroborate that the proposed method is suitable for both regular and irregular meshes regardless of geometrical complexity. Compared with the traditional variable mapping method, the proposed method needs less manual setup in the preprocessing of the analysis model for implementing extrusion constraint in topology optimization, which can be easily integrated into the solid isotropic materials with penalization (SIMP) topology optimization or commercial topology optimization software.

Short communication: Landlab v2.0: a software package for Earth surface dynamics

Abstract. Numerical simulation of the form and characteristics of Earth's surface provides insight into its evolution. Landlab is an open-source Python package that contains modularized elements of numerical models for Earth's surface, thus reducing time required for researchers to create new or reimplement existing models. Landlab contains a gridding engine which represents the model domain as a dual graph of structured quadrilaterals (e.g., raster) or irregular Voronoi polygon–Delaunay triangle mesh (e.g., regular hexagons, radially symmetric meshes, and fully irregular meshes). Landlab also contains components – modular implementations of single physical processes – and a suite of utilities that support numerical methods, input/output, and visualization. This contribution describes package development since version 1.0 and backward-compatibility-breaking changes that necessitate the new major release, version 2.0. Substantial changes include refactoring the grid, improving the component standard interface, dropping Python 2 support, and creating 31 new components – for a total of 58 components in the Landlab package. We describe reasons why many changes were made in order to provide insight for designers of future packages. We conclude by discussing lessons about the dynamics of scientific software development gained from the experience of using, developing, maintaining, and teaching with Landlab.

Hexafluoroisopropanol-alkanol based high-density supramolecular solvents: Fabrication, characterization and application potential as restricted access extractants

Hexafluoroisopropanol-alkanol (C6–C12) supramolecular solvents were proposed and their application potential as restricted access extraction solvents was assessed. Hexafluoroisopropanol acts as alkanol reverse micelle-forming agent and density-regulating agent for the supramolecular solvent synthesis. The formation of supramolecular solvent only needs a small percentage of hexafluoroisopropanol (<10% v/v for alkanol <5% v/v) and is almost not affected by the pH (2-11) and low ionic strength (NaCl <3%, w/v) of aqueous medium. The supramolecular solvents have higher density than water and consist of reversed micellar aggregates with irregular mesh structures and hydrophilic inner cavities, and the hydrophilic cavity size is obviously dependent on the hexafluoroisopropanol amount in the bulk system but almost not relevant to the alkanol amount. The hydrogen-bond and hydrophobic interactions of hexafluoroisopropanol with alkanol and the hydrogen-bond force between hexafluoroisopropanol and water play crucial roles in the formation of supramolecular solvent. The supramolecular solvents not only do not or slightly dissolve macromolecules (proteins, polysaccharides, and humic acid sodium), but also can effectively exclude them in extraction procedure; meanwhile, they can extract small molecules from complex samples with high extraction rate (even near 100% through optimization). The extraction mechanism involves the hydrophilic inner cavity size of supramolecular aggregates and the interactions (hydrogen-bond, hydrophobic) between supramolecular solvent and analytes. The suitability of supramolecular solvents as extractants was evaluated by extracting and determining chlorophenols in real water samples, coupled with high performance liquid chromatography-ultraviolet detector. The enrichment factors of 72–147 were obtained. Limits of detection were from 0.38 to 0.57 ng mL−1. The recoveries ranged from 96.0 to 107.9%, and intra-day and inter-day precisions were less than 4.0%. The novel hexafluoroisopropanol-alkanol supramolecular solvents have promising potential as restricted access extractants.

Traveltime computation and OBN seismic record simulation for rugged seafloor in VTI medium

ABSTRACT To improve the computational accuracy of seismic traveltime for complex structures in inhomogeneous media, a mesh generation method that is adaptive to a rugged seafloor or interface is presented. A hybrid mesh discrete velocity model consisting of conventional rectangular mesh and irregular quadrilateral mesh is also proposed. The local traveltime equations for the hybrid mesh are derived from the linear traveltime interpolation (LTI) method and are shown to be stable. The relationship between the group velocity and group angle is converted into that between group velocity and interpolated point coordinates, extending the LTI ray tracing algorithm to multiwave simulation in vertical transversely isotropic (VTI) medium by multi-stage partitioning. Computations of the first arrival, reflection, multiple reflection, multiple transmission conversion, and multiple reflection conversion seismic waves in the VTI medium with rugged seafloor and complex structural interfaces are realised. The results show that the proposed method can adapt to the rugged seafloor and velocity interfaces of complex structures, resulting in higher computational accuracy of the traveltime and ray paths for ocean bottom node (OBN) seismic wave simulation. The accuracy of the proposed method is verified by comparison with the finite-difference method. This method will be useful for modelling of seismic waves, wavefield identification, and study of seismic wave propagation in OBN seismic observations.

CSIOR: Circle-Surface Intersection Ordered Resampling

Triangular mesh is one of the most popular 3D modalities, which usage spans a wide variety of applications in computer vision, computer graphics and multimedia. Raw mesh surfaces generated by 3D scanning devices often suffer from mesh irregularity and also lack the implicit ordered structure that characterizes 2D images. Therefore, they are not suitable to be processed as such, especially for applications requiring fine surface analysis tasks. In this paper, we propose CSIOR, a novel mesh regularization technique exhibiting novel capabilities including (a) quasi-equilateral triangle and ordered tessellation that can accommodate non-convex shaped surfaces, (b) implicit generation of iso-geodesic contours, and (c) preservation of the object geometric properties, such as shape and surface corrugations (relief patterns). We evidence the superiority of our technique over current methods through a series of experiments performed on a variety of surfaces with relief patterns.

Positive Definite Advection Transport Algorithm for Conservation Law Equations on Nonuniform Irregular Grids

The multidimensional positive definite advection transport algorithm (MPDATA) is an important numerical method for the computation of atmospheric dynamics. MPDATA is second-order accurate, positive definite, conservative, and computationally efficient. However, the method is problematic in which it results in a loss of precision when computing a nonuniform irregular grid. Furthermore, research revealed two reasons for this problem. On the one hand, numerical discretization of boundary derivatives of the finite-volume method is incompatible with nonuniform meshes (or grids); on the other hand, the up-wind scheme of staggered grids is not applicable to the calculation of irregular grids. We overcome these two problems by using the multipoint Taylor expansion method to obtain a boundary derivative numerical approximation scheme that does not depend on the grid structure. Furthermore, combined with the well-balance central-upwind scheme, a positive definite advection scheme for irregular meshes is proposed. Then, the positivity of the new numerical scheme is analyzed. Finally, the result of this study is verified by numerical simulation.

EFFECTS OF SPATIAL VARIABILITY OF SOIL CHEMICAL PARAMETERS ON TIFTON 85 GRASS YIELD1

ABSTRACT Studies on spatial variability of soil attributes of tropical pastures gather information that can assist in decision making about managements of these soils. The objective of the present study was to evaluate the spatial variability of soil chemical attributes and their effects on grass yield of Tifton 85. The experiment was carried out in an area of 3.91 ha at the Feno Rio Farm of the Federal Rural University of Rio de Janeiro, Seropédica, RJ, Brazil. Soils of the 0-0.20 and 0.20-0.40 m layers were sampled considering an irregular sampling mesh, making a total of 50 georeferenced points. The parameters evaluated were: the soil chemical attributes pH, Al+3, Ca+2, Mg+2, Na+, K+, P, H+Al, and total organic carbon (TOC); and the Tifton 85 dry matter yield (DMY). The results of these parameters were subjected to descriptive statistics, linear correlation, and geostatistics, and maps were developed for the analyses. Regions with grass yields different from the general mean were found in the area, which presented mean grass yield of 2248 kg ha-1. The soil chemical parameters Na+, Ca+2, TOC, and H+Al were significantly correlated with DMY, confirming that they are important and affect the Tifton 85 grass yield. The mapping of the Tifton 85 cycle is important for understanding the variability of DMY. The investigation of areas with different productive potentials should be followed by development of maps of soil chemical attributes to correlate and understand the ratios that may be involved with these variations.

Isogeometric analysis of thin Reissner–Mindlin shells: locking phenomena and B-bar method

We propose a local type of B-bar formulation, addressing locking in degenerated Reissner–Mindlin shell formulation in the context of isogeometric analysis. Parasitic strain components are projected onto the physical space locally, i.e. at the element level, using a least-squares approach. The formulation allows the flexible utilization of basis functions of different orders as the projection bases. The introduced formulation is much cheaper computationally than the classical $$\bar{B}$$ method. We show the numerical consistency of the scheme through numerical examples, moreover they show that the proposed formulation alleviates locking and yields good accuracy even for slenderness ratios of $$10^5$$ , and has the ability to capture deformations of thin shells using relatively coarse meshes. In addition it can be opined that the proposed method is less sensitive to locking with irregular meshes.

Electromagnetic device modeling using a new adaptive wavelet finite element method

Wavelets provide sparse representations of integral and differential operators through a multiresolution approximation. This property is well-suited for adaptive implementation of numerical techniques, which is a common topic among researchers nowadays. The current state of the wavelet theory also allows flexible construction of transforms customized to certain applications, which is an important feature considering that the generalization of wavelet techniques to a large class of real-world numerical simulations is still limited. In this paper, it is proposed an adaptive wavelet finite element method for achieving operator-orthogonality between approximation spaces, a property which eliminates coupling among levels of resolution – a common shortcoming of adaptive methods. The resulting method is easily extended to any given finite element application and requires less computation power when compared to other adaptive techniques. These features are showcased by comparing the proposed technique to the popular h-refinement method using electromagnetic device simulations involving Dirichlet and Neumann boundary conditions, inhomogeneous and non-linear media, and irregular meshes.

Method for Analytically Finding the Nullspace of Stiffness Matrix for Both Zeroth-Order and Higher Order Curl-Conforming Vector Bases in Unstructured Meshes

The stiffness matrix in a vector finite-element method has a nullspace whose size grows linearly with the matrix size. Finding it numerically can be computationally expensive. However, the nullspace is important in many applications from solving the low-frequency breakdown problem to developing fast solvers. In this article, an analytical method to generate the nullspace of the stiffness matrix is developed, not only for the zeroth-order but also for any higher order vector basis function in arbitrary unstructured meshes. Using the mesh information, we are able to analytically construct the nullspace without any difficulty, thus avoiding solving an eigenvalue problem for finding the nullspace. The proposed analytical method has been applied to a variety of 2-D and 3-D irregular meshes for various orders of vector basis functions. Comparisons with the reference nullspace obtained from a brute-force eigenvalue solution have validated the proposed analytical method. The proposed work has also been applied to efficiently solve the low-frequency breakdown problem in a full-wave solver and perform fast layout parasitics extraction.

Virtual element method for the numerical simulation of long-term dynamics of transitional environments

The prediction of long-term dynamics of transitional environments, e.g., lagoon evolution, salt-marsh growth or river delta progradation, is an important issue to estimate the potential impacts of different scenarios on such vulnerable intertidal morphologies. The numerical simulation of the combined accretion and consolidation, i.e., the two main processes driving the dynamics of these environments, however, suffers from a significant geometric non-linearity, which may result in a pronounced grid distortion using standard grid-based discretization methods. The present work describes a novel numerical approach, based on the Virtual Element Method (VEM), for the long-term simulation of the vertical dynamics of transitional landforms. The VEM is a grid-based variational technique for the numerical discretization of Partial Differential Equations (PDEs) allowing for the use of very irregular meshes consisting of a free combination of different polyhedral elements. The model solves the groundwater flow equation, coupled to a geomechanical module based on Terzaghi's principle, in a large-deformation setting, taking into account both the geometric and the material non-linearity. The use of the VEM allows for a great flexibility in the element generation and management, avoiding the numerical issues connected with the adoption of strongly distorted meshes. The numerical model is developed, implemented and tested in real-world examples, showing an interesting potential for addressing complex environmental situations.

Structural joint inversion on irregular meshes

SUMMARY Structural joint inversion of several data sets on an irregular mesh requires appropriate coupling operators. To date, joint inversion algorithms are primarily designed for the use on regular rectilinear grids and impose structural similarity in the direct neighbourhood of a cell only. We introduce a novel scheme for calculating cross-gradient operators based on a correlation model that allows to define the operator size by imposing physical length scales. We demonstrate that the proposed cross-gradient operators are largely decoupled from the discretization of the modelling domain, which is particularly important for irregular meshes where cell sizes vary. Our structural joint inversion algorithm is applied to a synthetic electrical resistivity tomography and ground penetrating radar 3-D cross-well experiment aiming at imaging two anomalous bodies and extracting the parameter distribution of the geostatistical background models. For both tasks, joint inversion produced superior results compared with individual inversions of the two data sets. Finally, we applied structural joint inversion to two field data sets recorded over a karstified limestone area. By including geological a priori information via the correlation-based operators into the joint inversion, we find P-wave velocity and electrical resistivity tomograms that are in accordance with the expected subsurface geology.

Fractal Interpolation and Integration over Two-Dimensional Triangular Meshes

The fractal interpolation techniques are powerful alternatives to classical interpolation methods in case of complex irregular data such as financial time series, seismic data and biological signals. The main goal of this study is to apply fractal interpolation techniques over two-dimensional irregular meshes. We partition the polygonal interpolation domain into triangular regions and consider affine contraction mappings from triangular regions to triangular patches of the mesh. Calculation of the fractal interpolation coefficients is reduced to solving a linear system of equations. Freely chosen scaling variables of the iterated function system transformations provide flexibility. Additionally, the constructed iterated function system coefficients for the interpolation can be used directly to evaluate two-dimensional numerical integrals over the domain. We provide basic error results via numerical simulations.

NtkFoam: An OpenFOAM based neutron transport kinetics solver for nuclear reactor simulation

Due to the complexity of detailed nuclear reactor numerical simulation, especially the complicated geometry and multi-physics coupling properties of the advanced reactor, a nuclear reactor kinetics solver for nuclear reactor engineering and design needs to be developed. Based on the open source C++ software OpenFOAM, this work establishes a neutron transport kinetics solver, namely ntkFoam, for the nuclear reactor kinetics simulation, from the governing equations to the detailed implementations. The coupling between multi-group neutron transport equations and the delayed neutron precursor balance equations are considered with using the finite volume method. By introducing the inverse power method to the OpenFOAM, the k-eigenvalue problems are calculated, and by coupling the neutron transport and delayed neutron precursor calculations using the Euler implicit scheme, the transient kinetics problems are simulated. Numerical results show that the proposed ntkFoam can simulate the multi-group neutron transport kinetics problem accurately and flexibility, and both the regular and irregular mesh configurations can be adopted. This work can provide some new perspectives and foundations to nuclear reactor coupling calculations.

A greedy region growing algorithm for anisotropic stretch adaptive triangulation of geometry images

The geometry image representation is a remeshing of an irregular triangle surface mesh onto a rectangular grid of points and facilitates the processing of surface geometry. Its connectivity is commonly generated by splitting each rectangular quad of grid points along the shorter diagonal. However, this simple approach typically yields skinny triangles on the surface when the underlying local surface metric tensor eigenvalues differ significantly. Sequences of such triangles can appear like jaggedness.This paper presents a greedy, region growing algorithm for anisotropic triangulation of geometry images obtained by geometric stretch parametrization. The algorithm compensates for the local stretch anisotropy and variations in the principal directions of the metric tensor by minimizing the total length of the new edges of triangles added to the grown region. The surface reconstructed is more faithful to the original surface than the ones reconstructed by quad-splitting connectivity and recognized triangulation approaches for point clouds.

Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes

We consider numerical approximation of the degenerate advection-diffusion equation, which is formally parabolic but may exhibit hyperbolic behavior. We develop both explicit and implicit finite volume weighted essentially non-oscillatory (WENO) schemes in multiple space dimensions on non-uniform computational meshes. The diffusion degeneracy is reformulated through the use of the Kirchhoff transformation. Space is discretized using WENO reconstructions with adaptive order (WENO-AO), which have several advantages, including the avoidance of negative linear weights and the ability to handle irregular computational meshes. A special two-stage WENO reconstruction procedure is developed to handle degenerate diffusion. Element averages of the solution are first reconstructed to give point values of the solution, and these point values are in turn used to reconstruct the Kirchhoff transform variable of the diffusive flux. Time is discretized using the method of lines and a Runge-Kutta time integrator. We use Strong Stability Preserving (SSP) Runge-Kutta methods for the explicit schemes, which have a severe parabolically scaled time step restriction to maintain stability. We also develop implicit Runge-Kutta methods. SSP methods are only conditionally stable, so we discuss the use of L-stable Runge-Kutta methods. We present in detail schemes that are third order in both space and time in one and two space dimensions using non-uniform meshes of intervals or quadrilaterals. Efficient implementation is described for computational meshes that are logically rectangular. Through a von Neumann (or Fourier mode) stability analysis, we show that smooth solutions to the linear problem are unconditionally L-stable on uniform computational meshes when using an implicit Radau IIA Runge-Kutta method. Computational results show the ability of the schemes to accurately approximate challenging test problems.