In this paper, we propose a nonlinear positivity-preserving finite volume element(FVE) scheme for anisotropic diffusion problems on quadrilateral meshes. Based on an overlapping dual partition, the one-sided flux is approximated by the iso-parametric bilinear element. A positivity-preserving nonlinear scheme with vertex-centered unknowns is obtained by a new two-point flux technique, which avoids the convex decomposition of co-normals and the introduction of intermediate unknowns. The existence of a solution is proved for this nonlinear system by applying the Brouwer's theorem. Numerical results show that the proposed positivity-preserving scheme is effective on distorted quadrilateral meshes and has approximate second-order accuracy for both isotropic and anisotropic diffusion problems. Moreover, the presented scheme is applied on an equilibrium radiation diffusion problem with discontinuous coefficients. Numerical results show that the new scheme is much more efficient than the standard FVE method.
We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which when differentiated provides the derivatives of the original function. The method generalises traditional finite difference methods to meshes of arbitrary topology in any number of dimensions for any order of derivative and accuracy. We demonstrate the accuracy of the numerical scheme using dual quadrilateral meshes and a refinement method based on subdivision …
The $L^2$-orthogonal projection $Π_h:L^2(Ω)\rightarrow\mathbb{V}_h$ onto a finite element (FE) space $\mathbb{V}_h$ is called $H^1$-stable iff $\|\nablaΠ_h u\|_{L^2(Ω)}\leq C\|u\|_{H^1(Ω)}$, for any $u\in H^1(Ω)$ with a positive constant $C\neq C(h)$ independent of the mesh size $h>0$. In this work, we discuss local criteria for the $H^1$-stability of adaptively refined meshes. We show that adaptive refinement strategies for quadrilateral meshes in 2D (Q-RG and Q-RB), introduced originally in Bank et al. 1982 and Kobbelt 1996, are $H^1$-stable for FE spaces of polynomial degree $p=2,\ldots,9$.
A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids
Easy to construct and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. Nonetheless, especially for hexahedral meshes, the construction of smooth and optimally convergent isogeometric analysis basis functions is still an open question. We introduce a simple partition of unity construction that yields smooth blended B-splines, referred to as SB-splines, on semi-structured quadrilateral and hexahedral meshes, namely on mostly structured meshes with a few sufficiently separate…
High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points. In this work, we extend entropy-stable Gauss collocation schemes t…
In this paper we introduce a $C^1$ spline space over mixed meshes composed of triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis. In this context, a mesh is considered to be a partition of a planar polygonal domain into triangles and/or quadrilaterals. The proposed space combines the Argyris triangle, cf. (Argyris, Fried, Scharpf; 1968), with the $C^1$ quadrilateral element introduced in (Brenner, Sung; 2005) and (Kapl, Sangalli, Takacs; 2019) for polynomial degrees $p\geq 5$. The space is assumed to be $C^2$ at all vertices and $C^1$ across edges, and the splines ar…
Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes
The Rachel and Selim Benin School of Computer Science and Engineering（Rachel和Selim Benin计算机科学与工程学）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Microsoft Corp.（微软公司）
In this paper, based on the idea of self-adjusting steepness based schemes[5], a two-dimensional calculation method of steepness parameter is proposed, and thus a two-dimensional self-adjusting steepness based limiter is constructed. With the application of such limiter to the over-intersection based remapping framework, a low dissipation remapping method has been proposed that can be applied to the existing ALE method.
Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-$C^1$ are biquadratic splines on fully unstructured quadrilateral meshes (without restrictio…
The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular, quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh elements always increases and the mean square error (M…
Standard low-order finite elements, which perform well for problems involving compressible elastic materials, are known to under-perform when nearly incompressible materials are involved, commonly exhibiting the locking phenomenon. Interior penalty (IP) discontinuous Galerkin methods have been shown to circumvent locking when simplicial elements are used. The same IP methods, however, result in locking on meshes of quadrilaterals. The authors have shown in earlier work that under-integration of specified terms in the IP formulation eliminates the locking problem for rectangular elements. Here …
Isogeometric analysis with $C^1$ functions on unstructured quadrilateral meshes
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）；Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, and IMATI-CNR（帕维亚大学F. Casorati数学系及中国科学院应用数学研究所）；Institute of Applied Geometry, Johannes Kepler University Linz（林茨约翰开普勒大学应用几何研究所）
Given a sequence of finite element spaces which form a de Rham sequence, we will construct a dual representation of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The matrix which converts primal representations to dual representations -- the Hodge matrix -- is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even…
Linear Kinematic Features (LKFs) are found everywhere in the Arctic sea ice cover. They are strongly localized deformations often associated with the formation of leads and pressure ridges. Viscous-plastic sea ice models start to generate LKFs at high spatial grid resolution, typically with a grid spacing below 5 km. Besides grid spacing, other aspects of a numerical implementation, such as discretization details, may affect the number and definition of simulated LKFs. To explore these effects, the solutions of sea ice models with different grid spacings, mesh types, and numerical discretizati…
Solution interpolation between deforming meshes is an important component for several applications in scientific computing, including indirect arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical solution of partial differential equations. In this paper, a high-order, conservative, and positivity-preserving interpolation scheme is developed based on the discontinuous Galerkin solution of a linear time-dependent equation on deforming meshes. The scheme works for bounded but otherwise arbitrary mesh deformation from the old mesh to the new one. The cost and positivity pres…
Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement
High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuo…
A simple and efficient interface-fitted mesh generation algorithm is developed in this paper. This algorithm can produce a local anisotropic fitting mixed mesh which consists of both triangles and quadrilaterals near the interface. A new finite element method is proposed for second order elliptic interface problems based on the resulting mesh. Optimal approximation capabilities on anisotropic elements are proved in both the $H^1$ and $L^2$ norms. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a multigrid method is presented t…
We describe an adaptive version of a method for generating valid naturally curved quadrilateral meshes. The method uses a guiding field, derived from the concept of a cross field, to create block decompositions of multiply connected two dimensional domains. The a priori curved quadrilateral blocks can be further split into a finer high-order mesh as needed. The guiding field is computed by a Laplace equation solver using a continuous Galerkin or discontinuous Galerkin spectral element formulation. This operation is aided by using $p$-adaptation to achieve faster convergence of the solution wit…
Embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes
University of Milano–Bicocca（米兰比可卡大学）；IMATI-CNR “Enrico Magenes”（IMATI-CNR“恩里科·马盖斯”研究所）；University of Pavia（帕维亚大学）；University of Vienna（维也纳大学）；Georg-August-Universität Göttingen（哥廷根乔治-奥古斯特大学）
An efficient and stable diffusion generated method for quadrilateral mesh generation in general domains
The Hong Kong Polytechnic University（香港理工大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）
We present a high-order entropy stable discontinuous Galerkin (ESDG) method for the two dimensional shallow water equations (SWE) on curved triangular meshes. The presented scheme preserves a semi-discrete entropy inequality and remains well-balanced for continuous bathymetry profiles. We provide numerical experiments which confirm the high-order accuracy and theoretical properties of the scheme, and compare the presented scheme to an entropy stable scheme based on simplicial summation-by-parts (SBP) finite difference operators. Finally, we report the computational performance of an implementa…
This paper develops the structure-preserving finite volume weighted essentially non-oscillatory (WENO) hybrid schemes for the shallow water equations under the arbitrary Lagrangian-Eulerian (ALE) framework, dubbed as ALE-WENO schemes. The WENO hybrid reconstruction is adopted on moving meshes, which distinguishes the smooth, non-smooth, and transition stencils by a simple smoothness detector. To maintain the positivity preserving and the well-balanced properties of the ALE-WENO schemes, we adapt the positivity preserving limiter and the well-balanced approaches on static meshes to moving meshe…
A new high order accurate semi-implicit space-time Discontinuous Galerkin method on staggered grids, for the simulation of viscous incompressible flows on two-dimensional domains is presented. The designed scheme is of the Arbitrary Lagrangian Eulerian type, which is suitable to work on fixed as well as on moving meshes. In our space-time formulation, by expressing the numerical solution in terms of piecewise space-time polynomials, an arbitrary high order of accuracy in time is achieved through a simple and efficient method of Picard iterations. For the dual mesh, the basis functions consist …
We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute crosses. Instead, we compute a high order accurate guiding field using a continuous Galerkin (CG) or discontinuous Galerkin (DG) spectral element method to solve a Laplace equation for each of the field variables using the open source code Nektar++. The spectral method provides spectral convergence and sub-element resolution of the fields. The DG approximation al…
