Non-planar Faces Research Articles

Spectral element method for 3-D controlled-source electromagnetic forward modelling using unstructured hexahedral meshes

SUMMARYFor forward modelling of realistic 3-D land-based controlled-source electromagnetic (EM) problems, we develop a parallel spectral element approach, blending the flexibility and versatility of the finite element method in using unstructured grids with the accuracy of the spectral method. Complex-shaped structures and topography are accommodated by using unstructured hexahedral meshes, in which the elements can have curved edges and non-planar faces. Our code is the first spectral element algorithm in EM geophysics that uses the total field formulation (here that of the electric field). Combining unstructured grids and a total field formulation provides advantages in dealing with topography, in particular, when the transmitter is located on rough surface topography. As a further improvement over existing spectral element methods, our approach does not only allow for arbitrary distributions of conductivity, but also of magnetic permeability and dielectric permittivity. The total electric field on the elements is expanded in terms of high-order Lagrangian interpolants, and element-wise integration in the weak form of the boundary value problem is accomplished by Gauss–Legendre–Lobatto quadrature. The resulting complex-valued linear system of equations is solved using the direct solver MUMPS, and, subsequently, the magnetic field is computed at the points of interest by Faraday’s law. Five numerical examples comprehensively study the benefits of this algorithm. Comparisons to semi-analytical and finite element results confirm accurate representation of the EM responses and indicate low dependency on mesh discretization for the spectral element method. A convergence study illuminates the relation between high order polynomial approximation and mesh size and their effects on accuracy and computational cost revealing that high-order approximation yields accurate modelling results for very coarse meshes but is accompanied by high computational cost. The presented numerical experiments give evidence that 2nd and 3rd degree polynomials in combination with moderately discretized meshes provide better trade-offs in terms of computational resources and accuracy than lowest and higher order spectral element methods. To our knowledge, our final example that includes pronounced surface topography and two geometrically complicated conductive anomalies represents the first successful attempt at using 2nd order hexahedral elements supporting curved edges and non-planar faces in controlled-source EM geophysics.

A flux correction for finite-volume discretizations: Achieving second-order accuracy on arbitrary polyhedral grids

In this paper, we propose a flux correction technique generally applicable to practical finite-volume discretizations of a single flux evaluation per face for achieving second-order accuracy on arbitrary polyhedral grids involving non-planar faces. The proposed technique is derived from the k-exact finite-volume discretization approach originally introduced by Brenner. We take it as a general methodology for constructing a second-order discretization on arbitrary polyhedral grids, and identify a term considered as missing in other practical finite-volume discretizations; it is thus considered as a reason for the loss of second-order accuracy in many of such methods: e.g., a cell-centered discretization loses second-order accuracy on grids with non-planar faces as typical in polyhedral grids. In particular, we write the term as a vector involving flux gradients and show that it can be easily implemented as a correction to a numerical flux at each face. Also, we will show that a consistent definition of a control volume is critically important for achieving second-order accuracy on general polyhedral grids. We then discuss its general applicability and impact on accuracy and efficiency with simple but illustrative examples: the edge-based solver is made to achieve faster iterative convergence on irregular tetrahedral grids with adjustable centroids while preserving second-order accuracy; both the edge-based and cell-centered discretizations are made second-order accurate on irregular prismatic grids having non-planar faces.

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input that connect the PQ strips. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling-based representations of developable surfaces while still obtaining developable, curvature-aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its isolines are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight isolines is non-linear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle-based interactive freeform modeling of developable shapes.

A least squares based diamond scheme for 3D heterogeneous and anisotropic diffusion problems on polyhedral meshes

We propose a diamond scheme for 3D heterogeneous and anisotropic problem using piecewise linear approximation and least squares method. Since the continuity of the solution and flux is treated appropriately, the scheme and the vertex interpolation algorithm allow arbitrary diffusion tensors. In addition, we employ an adaptive weighted coefficient in the least squares problem of the vertex interpolation using a novel way to evaluate the magnitude of the diffusion anisotropy. Consequently our vertex interpolation has better performance than the least squares interpolation in most tested cases. The presented scheme and vertex interpolation algorithm do not need edge information, which leads to less topological searches and simplifies the programming compared with other edge-based ones. By means of unified calculations of the geometric quantities, this scheme is suitable for meshes with nonplanar faces. Numerical experiments show that our scheme is linearity-preserving and achieves nearly optimal accuracy for the solution on general meshes. Moreover, the new vertex interpolation algorithm has an ideal performance for heterogeneous and highly anisotropic diffusion problems which are tough for some existing algorithms.

A new isosurface extraction method on arbitrary grids

The development of interface-capturing methods (such as level-set, phase-field or volume of fluid (VOF) methods) for arbitrary 3D grids has further highlighted the need for more accurate and efficient interface reconstruction procedures. In this work, we propose a new method for the extraction of isosurfaces on arbitrary polyhedra that can be used with advantage for this purpose. The isosurface is extracted from volume fractions by a general polygon tracing procedure, which is valid for convex or non-convex geometries, even with non-planar faces. The proposed method, which can be considered as an extension of the marching cubes technique, produces consistent results even for ambiguous situations in polyhedra of arbitrary shape. To show the reproducibility of the results presented in this work, we provide the open source library isoap, which has been developed to implement the proposed method and includes test programs to demonstrate the successful extraction of isosurfaces on several grids with polyhedral cells of different types. We present results obtained not only for isosurface extraction from discrete volume fractions resulting from a volume of fluid method, but also from data sets obtained from implicit mathematical functions and signed distances to scanned surfaces. The improvement provided by the proposed method for the extraction of isosurfaces in arbitrary grids will also be very useful in other fields, such as CFD visualization or medical imaging.

ВЕРИФИКАЦИЯ SN-МОДЕЛИРОВАНИЯ ПРОГРАММЫ FRIGATE НА СПЕЦИАЛЬНЫХ ТЕСТОВЫХ ЗАДАЧАХ

The main features of the SN-block of program FRIGATE aimed at optimizing the processes of calculated justification are considered in this paper. The original features of program FRIGATE are noted, in particular, the ability to perform calculations on grids composed of arbitrary hexahedrons (including those containing non-planar faces) with an arbitrary way of joining grid cells along their faces. The program implements a wide range of boundary conditions (reflection, periodicity, rotational symmetry, surface source), which allow to create economical grid approximations of reactors with a high level of spatial detail and high quality of the form of the vast majority of hexahedral grid cells. FRIGATE program allows to perform calculations with quadratures with local thickening of nodes to account for the contribution of the effect of radiation through channels, cracks and gaps in structures. Also in the program FRIGATE the ability is implemented to calculate the method of "bootstrapping" large models in case of lack of computer memory in the calculation of the problem. The results of verification possibilities SN-block of program FRIGATE on two special developed problems and benchmark-experiment of fast reactor JOYO are presented in this paper.

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

We propose a new nonlinear positivity‐preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex‐centered one where the edge‐centered, face‐centered, and cell‐centered unknowns are treated as auxiliary ones that can be computed by simple second‐order and positivity‐preserving interpolation algorithms. Different from most existing positivity‐preserving schemes, the presented scheme is based on a special nonlinear two‐point flux approximation that has a fixed stencil and does not require the convex decomposition of the co‐normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star‐shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so‐called numerical heat‐barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second‐order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.

A new polyhedral element for the analysis of hexahedral-dominant finite element models and its application to nonlinear solid mechanics problems

A hexahedral-dominant finite element mesh can be easily constructed by cutting regular hexahedral elements in a simple block with CAD surfaces representing outer surfaces of a geometric model. Polyhedral elements with straight edges but possibly non-planar faces are generated at the domain boundaries, while regular hexahedral elements remain in the interior region. Shape functions for polyhedral elements are derived from moving least square approximation based on a tetrahedral subdivision of polyhedral domains by a centroid-based subdivision technique. The polyhedral shape functions in this study have similar properties to conventional finite element shape functions in terms of continuity and completeness within elements, compatibility across inter-element boundaries and the Kronecker-delta property. Furthermore, the present approach using hexahedral-dominant meshes with polyhedral elements at domain boundaries is successfully applied to solve large deformation problems of hyperelastic and elastic–plastic materials.

Polyhedral elements using an edge-based smoothed finite element method for nonlinear elastic deformations of compressible and nearly incompressible materials

Polyhedral elements with an arbitrary number of nodes or non-planar faces, obtained with an edge-based smoothed finite element method, retain good geometric adaptability and accuracy in solution. This work is intended to extend the polyhedral elements to nonlinear elastic analysis with finite deformations. In order to overcome the volumetric locking problem, a smoothing domain-based selective smoothed finite element method scheme and a three-field-mixed cell-based smoothed finite element method with nodal cells were developed. Using several numerical examples, their performance and the accuracy of their solutions were examined, and their effectiveness for practical applications was demonstrated as well.

Boolean operations on arbitrary polygonal and polyhedral meshes

A linearithmic floating-point arithmetic algorithm designed for solving usual boolean operations (intersection, union, and difference) on arbitrary polygonal and polyhedral meshes is described in this paper. This method does not dis-feature the inputs which can be two volume meshes, two surface meshes or one of each. It provides conformal meshes upon exit. It can be used in many pre- and post-processing applications in computational physics (e.g. cut-cell volume mesh generation or conservative remapping). The core idea is to consider any configuration as a polygonal cloud. The polygons are first triangulated, the intersections are solved, the polyhedral cells are then reconstructed from the conformal triangles cloud and finally their triangular faces are re-aggregated to polygons. This approach offers great flexibility regarding the admissible topologies: non-planar faces, concave faces or cells and some non-manifoldness are handled. The algorithm is described in detail and some current results are shown.

A Discrete Parametrized Surface Theory in ℝ3

We propose a discrete surface theory in ℝ3 that unites the most prevalent versions of discrete special parametrizations. Our theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory. A quad net is a map from a strongly regular polytopal cell decomposition of a surface with all faces being quadrilaterals into ℝ3 with nonvanishing straight edges. A polytopal cell decomposition is strongly regular if each edge connects distinct vertices and meets at most two faces. Notice, in particular, that nonplanar faces are admissible. In discrete differential geometry, quad nets are understood as discretizations of parametrized surfaces [9, 10, 13]. In this agenda many classes of special surfaces have been discretized using algebro-geometric approaches for integrable geometry—originally using discrete analogues of soliton theory techniques (e.g., discrete Lax pairs and finite-gap integration [6]) to construct nets, but more recently using the notion of 3D consistency (reviewed in [10]). As in the smooth setting, these approaches have been successfully applied to space forms (see, e.g., [15–17, 22]). As an example consider the case of K-surfaces, i.e., surfaces of constant negative Gauß curvature. The integrability equations of classical surface theory are equivalent to the famous sine-Gordon equation [1, 20]. In an integrable discretization, the sine-Gordon equation becomes a finite difference equation for which integrability is encoded by a certain closing condition around a 3D cube. Both in the smooth and discrete settings, integrability is bound to specific choices of parameterizations, such as asymptotic line parametrizations for K-surfaces. In this way different classes of surfaces, such as minimal surfaces or surfaces of constant mean curvature (CMC), lead to different partial differential equations and give rise to different parametrizations. In the discrete case, this is reflected by developments that treat different special surfaces by disparate approaches. These integrable discretizations maintain characteristic properties of their smooth counterparts (e.g., the transformation theory of Darboux, Bäcklund, Bianchi, etc.). What has been lacking, however, is a unified discrete theory that lifts the restriction of special surface parametrizations. Indeed, different from the case of classical smooth surface theory, existing literature does not provide a general discrete theory for quad nets.

A 3D GCL compatible cell-centered Lagrangian scheme for solving gas dynamics equations

Solving the gas dynamics equations under the Lagrangian formalism enables to simulate complex flows with strong shock waves. This formulation is well suited to the simulation of multi-material compressible fluid flows such as those encountered in the domain of High Energy Density Physics (HEDP). These types of flows are characterized by complex 3D structures such as hydrodynamic instabilities (Richtmyer-Meshkov, Rayleigh-Taylor, etc.). Recently, the 3D extension of different Lagrangian schemes has been proposed and appears to be challenging. More precisely, the definition of the cell geometry in the 3D space through the treatment of its non-planar faces and the limiting of a reconstructed field in 3D in the case of a second-order extension are of great interest. This paper proposes two new methods to solve these problems. A systematic and symmetric geometrical decomposition of polyhedral cells is presented. This method enables to define a discrete divergence operator leading to the respect of the Geometric Conservation Law (GCL). Moreover, a multi-dimensional minmod limiter is proposed. This new limiter constructs, from nodal gradients, a cell gradient which enables to ensure the monotonicity of the numerical solution even in presence of strong discontinuity. These new ingredients are employed into a cell-centered Lagrangian scheme. Robustness and accuracy are assessed against various representative test cases.

A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh

AbstractHigh-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.

A General and Effective Numerical Integration Method to Evaluate Triple Integrals Using Generalized Gaussian Quadrature

A general and effective numerical integration formula to evaluate all triple integrals with finite limits is proposed in this paper. The formula is derived by transforming the domain of integration to a zero-one cube. The general derivation along with results over specific regions like cuboid, tetrahedron, prism, pyramid and few regions having planar and non-planar faces is provided. Numerical results also are tabulated to validate the formula.

Boolean Operations on Arbitrary Polyhedral Meshes

A floating-point arithmetic algorithm designed for solving usual boolean operations (intersection, union, and difference) on arbitrary polyhedral meshes is described in this paper. It can be used in many pre- and post-processing applications in computational physics (e.g. cut-cell volume mesh generation or high order conservative remapping). The method provides conformal polyhedral meshes upon exit. The core idea is to triangulate the polygons, solve the intersections at the triangular level, reconstruct the polyhedra from the cloud of conformal triangles and then re-aggregate their triangular faces to polygons. This approach offers a great flexibility regarding the admissible topologies: non-planar faces, concave faces or cells and some non-manifoldness are handled. The algorithm is described in details and some preliminary results are shown.

A 3-D Volume-of-Fluid advection method based on cell-vertex velocities for unstructured meshes

A new geometrical Volume-of-Fluid (VOF) method for capturing interfaces on three-dimensional (3-D) Cartesian and unstructured meshes is introduced. The method reconstructs interfaces as first- and second-order piecewise planar approximations (PLIC), and advects volumes in a single unsplit Lagrangian–Eulerian (LE) geometrical algorithm based on constructing flux polyhedrons by tracing back the Lagrangian trajectories of the cell-vertex velocities. In this way, the situations of overlapping between flux polyhedrons are minimized, consequently, the accuracy in the solution of the advection equation is improved by minimizing the creation of overshoots (volume fractions over one), undershoots (volume fractions below zero) and wisps (fluid in void regions or vice versa). However, if not treated carefully, the use of cell-vertex velocities may result in the construction of flux polyhedrons that contain nonplanar faces and that do not conserve volume. Therefore, this work explains in detail a set of geometric algorithms necessary to overcome these two drawbacks. In addition, the new VOF method is analyzed numerically on 3-D Cartesian and unstructured meshes, first, by reconstructing the interface of spherical geometries and, second, by evaluating the final advection result of a sphere placed in a rotation, shear and deformation field.

Accurate Cell-Centered Discretizations for Modeling Multiphase Flow in Porous Media on General Hexahedral and Simplicial Grids

Summary We introduce an accurate cell-centered method for modeling Darcy flow on general quadrilateral, hexahedral, and simplicial grids. We refer to these discretizations as the multipoint-flux mixed-finite-element (MFMFE) method. The MFMFE method is locally conservative with continuous fluxes and can be viewed within a variational framework as a mixed finite-element method with special approximating spaces and quadrature rules. We study two versions of the method: with a symmetric quadrature rule on smooth grids and a nonsymmetric quadrature rule on rough grids. The framework allows for handling hexahedral grids with nonplanar faces defined by trilinear mappings from the reference cube. Moreover, the MFMFE method allows for local elimination of the velocity, which leads to a cell-centered pressure system. Theoretical and numerical results demonstrate first-order convergence on rough grids. Second-order superconvergence is observed on smooth grids. We also discuss a new splitting scheme for modeling multiphase flows that can treat higher-order transport discretizations for saturations. We apply the MFMFE method to obtain physically consistent approximations to the velocity and a reference pressure on quadrilateral or hexahedral grids, and a discontinuous Galerkin method for saturations. For higher-order saturations, we propose an efficient post-processing technique that gives accurate velocities in the interior of the gridblocks. Computational results are provided for flow in highly heterogeneous reservoirs, including different capillary pressures arising from different rock types.

High aspect ratio grid effects on the accuracy of Navier–Stokes solutions on unstructured meshes

The method of manufactured solutions is used to evaluate the stability and accuracy of several unstructured discretization schemes in the presence of high-aspect ratio grids. Both inviscid and viscous discretizations are studied by systematically varying parameters such as the aspect ratio, mesh stretching, curvature, skewness, and non-planar faces. For the inviscid terms, gradient reconstruction based on cell least squares and nodal Green–Gauss are considered. In particular, high curvature grids are shown to lead to instabilities with both classes of schemes and a new Green–Gauss scheme based on nodal projection is developed that retains stability and formal second-order accuracy under all conditions. A further advantage of the nodal Green–Gauss schemes is that the nodal values can be used in the viscous discretization as well. All the viscous schemes tested here are demonstrated to preserve second-order accuracy. Finally, for three-dimensional meshes, triangulation of non-planar faces is found to be necessary to preserve second-order accuracy.

A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra

In this paper, we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell-centered finite difference scheme. A special non-symmetric quadrature rule is employed that yields a positive definite cell-centered system for the pressure by eliminating local velocities. The method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids, including hexahedra with non-planar faces. Theoretical and numerical results indicate first-order convergence for the pressure and face fluxes.

Mesh quality effects on the accuracy of CFD solutions on unstructured meshes

The order of accuracy and error magnitude of node- and cell-centered schemes are examined on representative unstructured meshes and flowfield solutions for computational fluid dynamics. Specifically, we investigate the properties of inviscid and viscous flux discretizations for isotropic and highly stretched meshes using the Method of Manufactured Solutions. Grid quality effects are studied by randomly perturbing the base meshes and cataloguing the error convergence as a function of grid size. For isotropic grids, node-centered approaches produce less error than cell-centered approaches. Moreover, a corrected node-centered scheme is shown to maintain third order accuracy for the inviscid terms on arbitrary triangular meshes. In contrast, for stretched meshes, cell-centered schemes are favored, with cell-centered prismatic approaches in particular showing the lowest levels of error. In three dimensions, simple flux integrations on non-planar control volume faces lead to first-order solution errors, while second-order accuracy is recovered by triangulation of the non-planar faces.