Divergence Operator Research Articles

Odd generalized Einstein metrics on Lie groups

Abstract An odd generalized metric $$E_{-}$$ E - on a Lie group G of dimension n is a left-invariant generalized metric on a Courant algebroid $$E_{H, F}$$ E H , F of type $$B_{n}$$ B n over G with left-invariant twisting forms $$H\in \Omega ^{3}(G)$$ H ∈ Ω 3 ( G ) and $$F\in \Omega ^{2}(G)$$ F ∈ Ω 2 ( G ) . Given an odd generalized metric $$E_{-}$$ E - on G we determine the affine space of left-invariant Levi-Civita generalized connections of $$E_{-}$$ E - . Given in addition a left-invariant divergence operator $$\delta $$ δ we show that there is a left-invariant Levi-Civita generalized connection of $$E_{-}$$ E - with divergence $$\delta $$ δ and we compute the corresponding Ricci tensor $$\textrm{Ric}^{\delta }$$ Ric δ of the pair $$(E_{-}, \delta )$$ ( E - , δ ) . The odd generalized metric $$E_{-}$$ E - is called odd generalized Einstein with divergence $$\delta $$ δ if $$\textrm{Ric}^{\delta }=0$$ Ric δ = 0 . As an application of our theory, we describe all odd generalized Einstein metrics of arbitrary left-invariant divergence on all 3-dimensional unimodular Lie groups.

An Entropy Dynamics Approach to Inferring Fractal-Order Complexity in the Electromagnetics of Solids.

A fractal-order entropy dynamics model is developed to create a modified form of Maxwell's time-dependent electromagnetic equations. The approach uses an information-theoretic method by combining Shannon's entropy with fractional moment constraints in time and space. Optimization of the cost function leads to a time-dependent Bayesian posterior density that is used to homogenize the electromagnetic fields. Self-consistency between maximizing entropy, inference of Bayesian posterior densities, and a fractal-order version of Maxwell's equations are developed. We first give a set of relationships for fractal derivative definitions and their relationship to divergence, curl, and Laplacian operators. The fractal-order entropy dynamic framework is then introduced to infer the Bayesian posterior and its application to modeling homogenized electromagnetic fields in solids. The results provide a methodology to help understand complexity from limited electromagnetic data using maximum entropy by formulating a fractal form of Maxwell's electromagnetic equations.

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration

In this paper, we discuss a von Mises plasticity model with nonlinear isotropic hardening assuming small strains in a plane strain example of internally pressurised thick-walled cylinder subjected to different loading conditions. The elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress exceeds the yield stress, corrections are made locally through a return mapping algorithm. We present a novel method that uses a Radial Basis Function-Finite Difference (RBF-FD) approach with Picard iteration to solve the system of nonlinear equations arising from plastic deformation. This technique eliminates the need to stabilise the divergence operator and avoids special positioning of the boundary nodes, while preserving the elegance of the meshless discretisation and avoiding the introduction of new parameters that would require tuning. The results of the proposed method are compared with analytical and Finite Element Method (FEM) solutions. The results show that the proposed method achieves comparable accuracy to FEM while offering significant advantages in the treatment of complex geometries without the need for conventional meshing or special treatment of boundary nodes or differential operators.

The qP- and qSV-wave separation in 2D vertically transversely isotropic media and their applications in elastic reverse time migration

Compressional and shear wave separation is an important step in anisotropic elastic reverse time migration (ERTM). With separated wavefields, we can generate images between different wave types and reveal more subsurface physical properties, as well as remove unwanted crosstalk and improve image quality. Traditional Helmholtz decomposition based on divergence and curl operations for isotropic media cannot be directly extended to vertically transversely isotropic (VTI) media. Wave-mode separation methods, similar to the nonstationary spatial filter and Poynting vector, cannot preserve the phase and amplitude information of the original coupled wavefield, thus limiting their application in ERTM. Currently, the anisotropic wavefield decomposition methods include the wavenumber-domain approach, the low-rank approximation, and other separation approaches based on different approximations, e.g., weak or elliptical anisotropy. These methods are either computationally intensive or involve large separation errors, especially in models with strong heterogeneities and anisotropy. The VTI wavefield separation operator is essentially defined in a mixed space-wavenumber domain. We introduce scalar operators to transfer operations from this mixed space-wavenumber domain to the space domain. The local wave propagation direction in the scalar operators is estimated using the Poynting vector, which is highly computationally efficient in the space domain. When we apply the wave separation method to ERTM, we not only obtain the vectorized qP and qSV waves but also retrieve the scalar qP wavefield. The scalar and vector wavefields preserve the amplitude and phase information in the original coupled wavefield. For anisotropic ERTM, we suggest using the scalar imaging condition to generate the PP image and the magnitude- and sign-based vector imaging condition to produce the PS image, both having higher image accuracy than the dot-product imaging condition. Numerical examples are used to validate our anisotropic wavefield separation method and the related ERTM workflow.

Markov generators as non-Hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and singular value decompositions

Continuity equations associated with continuous-time Markov processes can be considered as Euclidean Schrödinger equations, where the non-Hermitian quantum Hamiltonian H=divJ is naturally factorized into the product of the divergence operator div and the current operator J . For non-equilibrium Markov jump processes in a space of N configurations with M links and C=M−(N−1)⩾1 independent cycles, this factorization of the N × N Hamiltonian H=I†J involves the incidence matrix I and the current matrix J of size M × N, so that the supersymmetric partner H^=JI† governing the dynamics of the currents existing on the M links is of size M × M. To better understand the relations between the spectral decompositions of these two Hamiltonians H=I†J and H^=JI† with respect to their bi-orthogonal basis of right and left eigenvectors that characterize the relaxation dynamics towards the steady state and the steady currents, it is useful to analyze the properties of the singular value decomposition of the two rectangular matrices I and J of size M × N and the interpretations in terms of discrete Helmholtz decompositions. This general framework concerning Markov jump processes can be adapted to non-equilibrium diffusion processes governed by Fokker–Planck equations in dimension d, where the number N of configurations, the number M of links and the number C=M−(N−1) of independent cycles become infinite, while the two matrices I and J become first-order differential operators acting on scalar functions to produce vector fields.

Gradient discretisation of elliptic quasi-variational inequalities with gradient type barriers

AbstractThis paper proposes a discretisation scheme for quasi-variational inequalities problems with different constraint types associated with gradient and divergence operators. We use the gradient discretisation method, a generic framework designed to encompass several numerical schemes. The paper describes the discretisation and analyses the convergence using a compactness argument that does not require strong assumptions on a solution.

A new class of efficient high order semi-Lagrangian IMEX discontinuous Galerkin methods on staggered unstructured meshes

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

OPERATOR-DIFFERENCE APPROXIMATIONS ON NON-STANDARD RECTANGULAR GRIDS

In the approximate solution of boundary value problems for partial differential equations, difference methods are widely used. Grid approximations are most simply constructed when dividing the calculated area into rectangular cells. Usually the grid nodes coincide with the vertices of the cells. In addition to such nodal approximations, grids with nodes in the centers of cells are also used. It is convenient to formulate boundary value problems in terms of invariant operators of vector (tensor) analysis, which are compared with the corresponding grid analogues. The paper builds analogues of gradient and divergence operators on non-standard rectangular grids, the nodes of which consist of both the vertices of the calculated cells and their centers. The proposed approach is illustrated by approximations of the boundary value problem for the stationary two-dimensional convection-diffusion equation. The key features of the construction of approximations for vector problems with orientation to applied problems of solid mechanics are noted.

Flexibilities and Constraints – Confessionalism and the New Communities in Lebanon

The paper analyses the Lebanese political system called Lebanese consociationalism (confessionalism) focusing on its dynamics and reaction to social changes, particularly demographic. Scholarly literature defines the system as a power-sharing model, a set of political tools implemented to contain challenges arising from ethno-religious pluralism of a society. Important section of the theory is dedicated to conditions that impact dynamics of these systems and decide about its reaction towards challenges, in consequence impacting their stability, durability and success. The following paper joins discussion about Lebanese confessionalism and its reaction towards challenges, especially those posed by changing demographic composition. The paper refers first to the case of the Armenian community in Lebanon and their successful integration into the system also labelled as „lebanonization” as a significant example of the system’s dynamics. As contrasting case the paper also recalls the case of total systemic rejection and marginalization of a community – the Palestinians who came to Lebanon after 1948. The aim is to point at the conditions influencing the system’s divergent responses and operation in the context of dealing with the new communities.

Lighter and faster simulations on domains with symmetries

A strategy to improve the performance and reduce the memory footprint of simulations on meshes with spatial reflection symmetries is presented in this work. By using an appropriate mirrored ordering of the unknowns, discrete partial differential operators are represented by matrices with a regular block structure that allows replacing the standard sparse matrix–vector product with a specialised version of the sparse matrix-matrix product, which has a significantly higher arithmetic intensity. Consequently, matrix multiplications are accelerated, whereas their memory footprint is reduced, making massive simulations more affordable. As an example of practical application, we consider the numerical simulation of turbulent incompressible flows using a low-dissipation discretisation on unstructured collocated grids. All the required matrices are classified into three sparsity patterns that correspond to the discrete Laplacian, gradient, and divergence operators. Therefore, the above-mentioned benefits of exploiting spatial reflection symmetries are tested for these three matrices on both CPU and GPU, showing up to 5.0x speed-ups and 8.0x memory savings. Finally, a roofline performance analysis of the symmetry-aware sparse matrix–vector product is presented.

Sharp affine weighted L 2 Sobolev inequalities on the upper half space

Abstract We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.

Surface charge and surface current densities at material boundaries

In electromagnetism, materials with a polarization density P→ or a magnetization density M→ are known to exhibit a bound surface charge density σb=P→·n̂ or a surface current density κ→b=M→×n̂, respectively, where n̂ is the unit vector perpendicular to the material boundary surface, directed outward. These expressions can be obtained from volume integrations for the electric potential V, or the magnetic vector potential A→, in which the integrals are restricted to the material volumes delimited by their respective boundaries. In that case, applying the divergence theorem leads to surface integrals on material boundaries and to the above-mentioned surface quantities. In this paper, a simple derivation is presented, which shows that both σb and κ→b are included in the expressions for the volume charge or current densities, provided that the divergence and curl operators are evaluated at the boundary so as to account for discontinuities at interfaces.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

Isotropic discretization methods of Laplacian and generalized divergence operators in phase field models

Phase field models have been extensively used to address various physical phenomena. However, discretization-induced anisotropy remains a longstanding challenge for phase field models. By using a hexagonal mesh in 2D, we describe isotropic discretization methods for the computation of Laplacian and generalized divergence operators. Quantitative analyses derived from discrete Fourier analysis prove that our methods are more isotropic than commonly used approaches, including isotropic methods for square mesh. To compare the performance of conventional and our discretization methods, a specific phase field model of alloy solidification was selected to perform benchmark simulations. Various 2D simulations using different discretization methods were carried out to verify the accuracy and efficiency of the improved numerical methods in a hexagonal mesh. We emphasize that the improved numerical methods using a hexagonal mesh are general and may be equally applied to other physical models that include the same operators.

Divergent Synthesis of 3-(Indol-2-yl)quinoxalin-2-ones and 4-(Benzimidazol-2-yl)-3-methyl(aryl)cinnolines via Polyphosphoric Acid (PPA)-Mediated Intramolecular Rearrangements of 3-(Methyl/aryl(2-phenylhydrazono)methyl)quinoxalin-2-ones.

Herein, we report a polyphosphoric acid (PPA)-mediated divergent metal-free operation to access a diverse collection of 3-(indol-2-yl)quinoxalin-2-ones and 4-(benzimidazol-2-yl)-3-methylcinnolines in moderate to excellent overall yields. The described process involves two distinct, and competing rearrangements of 3-(methyl(2-phenylhydrazono)methyl)quinoxalin-2-ones, namely [3,3]-sigmatropic Fischer rearrangement with the formation of an indole ring to produce 3-(indol-2-yl)-quinoxalin-2-ones, and Mamedov rearrangement with simultaneous construction of benzimidazole and cinnoline rings to form the new biheterocyclic system─4-(benzimidazol-2-yl)-3-methylcinnolines. The reaction mechanism of both rearrangement channels is explored by extensive dispersion-corrected DFT calculations. It is partcularly remarkable that when 3-(aryl(2-phenylhydrazono)methyl)quinoxalin-2-ones is used, instead of 3-(methyl(2-phenylhydrazono)methyl)quinoxalin-2-ones, reactions proceed regioselectively with the formation of only rearrangement products─4-(benzimidazol-2-yl)-3-arylcinnolines with high yields. This operationally simple protocol enables a rapid access to these scaffolds and is compatible with a wide scope of starting materials. In addition, the new rearrangement found features a promising approach for the design of unique compound libraries for drug design and discovery programs.

A hybrid radial basis function-finite difference method for modelling two-dimensional thermo-elasto-plasticity, Part 1: Method formulation and testing

A hybrid version of the strong form meshless Radial Basis Function-Finite Difference (RBF-FD) method is introduced for solving thermo-mechanics. The thermal model is spatially discretised with RBF-FD, where trial functions are polyharmonic splines augmented with polynomials. For time discretisation, the explicit Euler method is employed. An extension of RBF-FD, the hybrid RBF-FD, is introduced for solving mechanical problems. The model is one-way coupled, where temperature affects displacements. The thermo-elastoplastic material response is considered where the stress field is generally non-smooth. The hybrid RBF-FD, where the finite difference method is used to discretise the divergence operator from the balance equation, is shown to be successful when dealing with such problems. The mechanical model is introduced in a plane strain and in a generalised plane strain (GPS) assumption. For the first time, this work presents a strong form RBF-FD for GPS problems subjected to integral form constraints. The proposed method is assessed regarding h-convergence and accuracy on the benchmark with heating an elastoplastic square. It is proven to be successful at solving one-way coupled thermo-elastoplastic problems. The proposed novel meshless approach is efficient, accurate, and robust. Its use in an industrial situation is provided in Part 2 of this paper.

A WENO SPH scheme with improved transport velocity and consistent divergence operator

The Arbitrary Lagrangian–Eulerian Smoothed Particle Hydrodynamics (ALE-SPH) formulation can guarantee stable solutions preventing the adoption of empirical parameters such as artificial viscosity. However, the convergence rate of the ALE-SPH formulation is still limited by the inaccuracy of the SPH spatial operators. In this work, a Weighted Essentially Non-Oscillatory (WENO) spatial reconstruction is then adopted to minimise the numerical diffusion introduced by the approximate Riemann solver (which ensures stability), in combination with two alternative approaches to restore the consistency of the scheme: corrected divergence SPH operators and the particle regularisation guaranteed by the correction of the transport velocity. The present work has been developed in the framework of the DualSPHysics open-source code. The beneficial effect of the WENO reconstruction to reduce numerical diffusion in ALE-SPH schemes is first confirmed by analysing the propagation of a small pressure perturbation in a fluid initially at rest. With the aid of a 2-D vortex test case, it is then demonstrated that the two aforementioned techniques to restore consistency effectively reduce saturation in the convergence to the analytical solution. Moreover, high-order (above second) convergence is achieved. Yet, the presented scheme is tested by means of a circular blast wave problem to demonstrate that the restoration of consistency is a key feature to guarantee accuracy even in the presence of a discontinuous pressure field. Finally, a standing wave has been reproduced with the aim of assessing the capability of the proposed approach to simulate free-surface flows.

Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains.

An improved local radial basis function method for solving small-strain elasto-plasticity

Strong-form meshless methods received much attention in recent years and are being extensively researched and applied to a wide range of problems in science and engineering. However, the solution of elasto-plastic problems has proven to be elusive because of often non-smooth constitutive relations between stress and strain. The novelty in tackling them is the introduction of virtual finite difference stencils to formulate a hybrid radial basis function generated finite difference (RBF-FD) method, which is used to solve small-strain von Mises elasto-plasticity for the first time by this original approach. The paper further contrasts the new method to two alternative legacy RBF-FD approaches, which fail when applied to this class of problems. The three approaches differ in the discretization of the divergence operator found in the balance equation that acts on the non-smooth stress field. Additionally, an innovative stabilization technique is employed to stabilize boundary conditions and is shown to be essential for any of the approaches to converge successfully. Approaches are assessed on elastic and elasto-plastic benchmarks where admissible ranges of newly introduced free parameters are studied regarding stability, accuracy, and convergence rate.