Divergence Theorem Research Articles

The Caccioppoli ultrafunctions

AbstractUltrafunctions are a particular class of functions defined on a hyperreal fieldℝ∗⊃ℝ{\mathbb{R}^{\ast}\supset\mathbb{R}}. They have been introduced and studied in some previous works [2, 6, 7]. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.

A divergence theorem for pseudo-Finsler spaces

We study the divergence theorem on pseudo-Finsler spaces and obtain a completely Finslerian version for spaces having a vanishing mean Cartan torsion. This result helps to clarify the problem of energy-momentum conservation in Finsler gravity theories.

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme and the least-squares (LS) scheme. Both are widely believed to be second-order accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain partial differential equation solver OpenFOAM.

A linear smoothed higher-order CS-FEM for the analysis of notched laminated composites

Higher-order elements with highly accurate solutions are attractive for stress analysis and stress concentration problems. However, the distorted eight-node serendipity quadrilateral element is known to yield inaccurate results and sub-optimal convergence rate. In this paper, we present a higher-order CS-FEM to alleviate the effect of distorted mesh and guarantee the quality of solutions by employing a linear smoothing technique over eight-node quadratic serendipity elements. The modified strain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor's expansion of the weak form. The proposed method eliminates the need for isoparametric mapping and numerical studies demonstrate that the proposed method is insensitive to mesh distortion. The improved accuracy and superior convergence rates are numerically demonstrated with a few benchmark problems. The analysis of the stress concentration around cutouts also proves that the present method has good performance for the laminated composites.

Analytical integration of 0th, 2nd, and 4th order polynomial filtering functions on unstructured grid for dispersed phase fraction computation in an Euler–Lagrange approach

This paper presents an analytical approach to evaluate the volume integrals emerging during dispersed phase fraction computation in Lagrangian–Eulerian methods. It studies a zeroth, second, and fourth order polynomial filtering function in test cases featuring structured and unstructured grids. The analytical integration is enabled in three steps. First, the divergence theorem is applied to transform the volume integral into surface integrals over the volumes’ boundaries. Secondly, the surfaces are projected alongside the first divergence direction. Lastly, the divergence theorem is applied for the second time to transform the surface integrals into line integrals. We propose a generic strategy and simplifications to derive an analytical description of the complex geometrical entities such as non-planar surfaces. This strategy enables a closed solution to the line integrals for polynomial filtering functions. Furthermore, this paper shows that the proposed approach is suitable to handle unstructured grids. A sine wave and Gaussian filtering function is tested and the fourth order polynomial is found to be a good surrogate for the sine wave filtering function as no expensive trigonometric evaluations are necessary.

Application of Gauss's theorem to quantify localized surface emissions from airborne measurements of wind and trace gases

Abstract. Airborne estimates of greenhouse gas emissions are becoming more prevalent with the advent of rapid commercial development of trace gas instrumentation featuring increased measurement accuracy, precision, and frequency, and the swelling interest in the verification of current emission inventories. Multiple airborne studies have indicated that emission inventories may underestimate some hydrocarbon emission sources in US oil- and gas-producing basins. Consequently, a proper assessment of the accuracy of these airborne methods is crucial to interpreting the meaning of such discrepancies. We present a new method of sampling surface sources of any trace gas for which fast and precise measurements can be made and apply it to methane, ethane, and carbon dioxide on spatial scales of ∼ 1000 m, where consecutive loops are flown around a targeted source region at multiple altitudes. Using Reynolds decomposition for the scalar concentrations, along with Gauss's theorem, we show that the method accurately accounts for the smaller-scale turbulent dispersion of the local plume, which is often ignored in other average mass balance methods. With the help of large eddy simulations (LES) we further show how the circling radius can be optimized for the micrometeorological conditions encountered during any flight. Furthermore, by sampling controlled releases of methane and ethane on the ground we can ascertain that the accuracy of the method, in appropriate meteorological conditions, is often better than 10 %, with limits of detection below 5 kg h−1 for both methane and ethane. Because of the FAA-mandated minimum flight safe altitude of 150 m, placement of the aircraft is critical to preventing a large portion of the emission plume from flowing underneath the lowest aircraft sampling altitude, which is generally the leading source of uncertainty in these measurements. Finally, we show how the accuracy of the method is strongly dependent on the number of sampling loops and/or time spent sampling the source plume.

Totally Volume Integral of Fluxes for Discontinuous Galerkin Method (TVI-DG) II-One Dimensional System of Equations

This is the second paper in a series in which we construct the totally volume integrals of Riemann flux mimicking Godunov flux for discontinuous Galerkin method. In this work the boundaries integrals of the Riemann fluxes are transformed into volume integral. The new family of DG method is accomplished by applying the divergence theorem to the boundaries integrals of the Riemann fluxes. Therefore, the discontinuous Galerkin (DG) method is independent on the boundaries integrals of fluxes at the cell (element) boundaries as in the classical discontinuous Galerkin (DG) methods. The modified streamline upwind Petrov-Galerkin method is used to capture the oscillation of unphysical flow for shocked flow problems. The numerical results of applying totally volume integral discontinuous Galerkin method (TVI-DG) are presented for Euler's equations in one-dimensional cases. The numerical finding of this scheme is very accurate as compared to exact solutions.

A linear smoothed quadratic finite element for the analysis of laminated composite Reissner–Mindlin plates

It is well known that the high-order elements have significantly improved the accuracy of solutions in the traditional finite element analysis, but the performance of high-order elements is restricted by the shear-locking and distorted meshes for the plate problems. In this paper, a linear smoothed eight-node Reissner-Mindlin plate element (Q8 plate element) based on the first order shear deformation theory is developed for the static and free vibration analysis of laminated composite plates, the computation of the interior derivatives of shape function and isoparametric mapping can be removed. The strain matrices are modified with a linear smoothing technique by using the divergence theorem between the nodal shape functions and their derivatives in Taylor’s expansion. Moreover, the first order Taylor’s expansion is also employed for the construction of stiffness matrix to satisfy the linear strain distribution. Several numerical examples indicate that the novel Q8 plate element has good performance to alleviate the shear-locking phenomenon and improve the quality of the solutions with distorted meshes.

A Kronecker-type identity and the representations of a number as a sum of three squares

By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kronecker-type identity, we also deduce Gauss's theorem that every positive integer is representable as a sum of three triangular numbers.

Totally Volume Integral of Fluxes for Discontinuous Galerkin Method (TVI-DG) I-Unsteady Scalar One Dimensional Conservation Laws

The volume integral of Riemann flux in the discontinuous Galerkin (DG) method is introduced in this paper. The boundaries integrals of the fluxes (Riemann flux) are transformed into volume integral. The new family of DG method is accomplished by applying divergence theorem to the boundaries integrals of the flux. Therefore, the (DG) method is independent of the boundaries integrals of fluxes (Riemann flux) at the cell (element) boundaries as in classical (DG) methods. The modified streamline upwind Petrov-Galerkin method is used to capture the oscillation of unphysical flow for shocked flow problems. The numerical results of applying totally volume integral discontinuous Galerkin method (TVI-DG) are presented to unsteady scalar hyperbolic equations (linear convection equation, inviscid Burger's equation and Buckley-Leverett equation) for one dimensional case. The numerical finding of this scheme is very accurate as compared with other high order schemes as the weighted compact finite difference method WCOMP.

Hardy type inequalities on the sphere

In this paper, we consider the L^{p}-Hardy inequalities on the sphere. By the divergence theorem, we establish the L^{p}-Hardy inequalities on the sphere. Furthermore, we also obtain their best constants. Our results can be regarded as the extension of Xiao’s (J. Math. Inequal. 10:793-805, 2016).

Naturally stabilized nodal integration meshfree formulations for analysis of laminated composite and sandwich plates

This article addresses a naturally stabilized nodal integration (NSNI) meshfree formulation for static, free vibration and buckling analyses of laminated composite and sandwich plates based on the higher order shear deformation theory (HSDT). The crucial idea is to directly compute integrations at nodes similar to the direct nodal integration (DNI) but the instability existing in DNI is avoided and the computational cost is significantly reduced when compared to traditional high-order Gauss quadrature schemes. Due to the displacements, strains and stresses directly computed at nodes, the computing time of the present approach is equivalent with the DNI. Being different from the stabilized conforming nodal integration technique using a divergence theorem to evaluate gradient strains by the surface-to-boundary integration, NSNI is a naturally implicit gradient expansion. Variational consistency in Galerkin weak form is explicitly described. Thanks to the satisfied Kronecker delta function property, the moving Kriging integration (MKI) shape function is used to impose directly essential boundary conditions as same as the finite element method (FEM). A variety of numerical examples with various complex geometries, aspect ratios, stiffness ratios and boundary conditions are studied. Numerical results show the effectiveness of the present method in comparison with some existing ones in the literature.

On deriving the Maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields

Though extensively used in calculating optical force and torque acting on a material object illuminated by laser, the Maxwell stress tensor (MST) method follows the electromagnetic linear and angular momentum balance that is usually derived in most textbooks for a continuous volume charge distribution in free space, if not resorting to the application of Noether’s theorem in electrodynamics. To cast the conservation laws into a physically appealing form involving the current densities of linear and angular momentum, on which the MST method is based, the divergence theorem is employed to transform a volume integral into a surface integral. When a material object of finite volume is put into the field, it brings about a discontinuity of field across its surface, due to the presence of induced surface charge and surface current. Ambiguity arises among students in whether the divergence theorem can still be directly used without any justification. By taking into account the effect of the induced surface charge and current, we present a simple pedagogical derivation for the MST method for calculating the optical force and torque on an object immersed in monochromatic optical field, without resorting to Noether’s theorem. Although the results turn out to be identical to those given in the standard textbooks, our derivation avoids the direct use of the divergence theorem on a discontinuous function.

Evaluation of 4-D Reaction Integrals in the Method of Moments: Coplanar Element Case

Recently, the benefits of simultaneously treating source and testing integrals in the numerical evaluation of 4-D reaction integrals have been reported. The reported schemes usually first transform the reaction integral to parametric coordinates, and some combination of radial, angular, and/or line segment integrals is then used to treat coincident, edge-adjacent, or vertex-adjacent triangular source and test element pairs. However, advantages of the reported approaches are tempered by their lack of generality and severely degraded performance on poorly shaped elements, the latter caused primarily by the parametric transformations’ severe distortion of the kernel’s circularly concentric level contours. Here, for coplanar element pairs and kernels with 1/ $R$ singularities, we apply the surface divergence theorem twice to obtain a novel formula for 4-D reaction integrals, generalizing earlier schemes while retaining their benefits and without distorting the original configuration space. Numerical results illustrate the method’s efficiency, which is improved by employing appropriate transformations to further smooth the resulting integrands and hence accelerate their convergence. The reaction integral formula can be extended to noncoplanar elements.

Hardy spaces of Clifford algebra-valued monogenic functions in exterior domains and a higher dimensional version of Cauchy’s vanishing theorem

We introduce and study Hardy spaces consisting of Clifford algebra-valued functions annihilated by perturbed Dirac operators in exterior uniformly rectifiable (UR) domains, and which radiate at infinity. In this context, we establish a higher dimensional version of Cauchy’s vanishing theorem, whose proof makes use of the properties of Cauchy-like operators in exterior UR domains, a sharp version of the Divergence Theorem in exterior Ahlfors regular domains, and a good understanding of the nature of various radiation conditions and properties of the far field pattern for Clifford algebra-valued null-solutions of the Helmholtz operator.

Axillary nodal prognostic impact of two-dimensional tumor heterogeneity flux in breast cancer: evaluation via dynamic contrast-enhanced magnetic resonance imaging

Background: Breast cancer remains to have a high mortality toll in women worldwide. Several studies indicate benefit in identifying axillary nodal involvement towards effective patient risk stratification and treatment. Tumor heterogeneity recently emerged as a key factor impacting clinical outcomes. Voxel intensity gradient (VIG), a measure of contrast amongst radiological features of heterogeneity, harbors significance in determining intra-tumoral properties. Our project aims to apply the divergence theorem in correlating the 2-dimensional (2D) surface flux of this quantity to extra-tumoral axillary nodal involvement.

Nonlocal Force Equilibrium Condition for Non-Simple Materials

For simple materials Noll’s principle of local action yields the stress tensor function to depend only on the local deformation gradient or its history (Noll in The Foundations of Mechanics and Thermodynamics, 1974, p. 20, Theorem 3). Consequently, the stress field is of class \(C^{1}\) and the standard force equilibrium condition exhibits the divergence of stress. Nonlocal models, e.g., couple stress theories, drop the principle of local action. They account for higher gradients in deformation and additional kinematical variables, respectively. Then, the stress tensor field in the contiguity of a continuum point may not be a linear function. In the context of power series expansion, higher order terms of stress appear in the representative volume element around the point. We axiomatically consider the stress field tensor and the body force vector as nonlinear functions of class \(C^{n}\), approximated via power series expansion of order \(m \leq n\) from the midpoint of a cubic representative volume element. Depending on the grade of approximation, the series expansion reproduces nonlinearities of the stress field in the cube and also on its surface. The proposed procedure yields a nonlocal force equilibrium condition extending the local condition by an additional term with internal length scale parameter. It evolves from integrating tractions on the surface of a finite region. Thus, we make no use of Green’s divergence theorem. Our approach is not restricted by material constitution. Thus, it is valid for solids and fluids. However, we limit our examples to solids, where an internal length scale arises from the inner structure of material. Additionally, the variational approach of a gradient elasticity model with explicit constitutive assumptions is under investigation. Latter leads to a similar force equilibrium condition, however, with a reversed sign for the proposed extension.

On the quantisation of complex higher derivative theories and avoiding the Ostrogradsky ghost

Generic higher derivative theories are believed to be fundamentally unphysical because they contain Ostrogradsky ghosts. We show that within complex classical mechanics it is possible to construct higher derivative theories that circumvent the Ostrogradsky theorem and have a real energy spectrum that is bounded from below. The complex theory can be canonically quantised. The resulting quantum theory does not suffer from the kinetic instability and maintains the usual probabilistic interpretation without violating the correspondence principle. As a proof of concept, we construct a class of stable interacting complex higher derivative theories and present a concrete example. This consistent and canonical framework allows us to analyse the previous attempts to avoid ghosts that use non-canonical quantisation schemes, such as the Lee–Wick theories, Dirac–Pauli quantisation or PT-symmetric quantum mechanics. The key to understand the would-be ghosts in any kinetically stable higher derivative theory is to accept the complex system behind it.

Exact integration scheme for planewave-enriched partition of unity finite element method to solve the Helmholtz problem

In this paper, we present an exact integration scheme to compute highly oscillatory integrals that appear in the solution of the two-dimensional Helmholtz problem using the planewave-enriched partition of unity finite element method. In the proposed scheme, such oscillatory integrals are computed by a recursive application of the divergence theorem, eventually expressing the integrals in terms of evaluations of the corresponding integrands at the nodes of the finite element mesh. The number of such function evaluations is independent of the wave number k, which permits the scheme to be used for arbitrary high values of k. We consider finite element meshes with unstructured triangular and structured rectangular elements, and present numerical results for three canonical benchmark Helmholtz problems to demonstrate the accuracy and efficacy of the method.

Non Machinable Volume Calculation Method for 5-Axis Roughing Based on Faceted Models through Closed Bounded Area Evaluation

The increase in the volume of rough machining on the CBV area is one of the indicators of increased efficiencyof machining process. Normally, this area is not subject to the rough machining process, so that the volume of the rest of the material is still big. With the addition of CC point and tool orientation to CBV area on a complex surface, the finishing will be faster because the volume of the excess material on this process will be reduced. This paper presents a method for volume calculation of the parts which do not allow further occurrence of the machining process, particulary for rough machining on a complex object. By comparing the total volume of raw materials and machining area volume, the volume of residual material,on which machining process cannot be done,can be determined. The volume of the total machining area has been taken into account for machiningof the CBV and non CBV areas. By using delaunay triangulation for the triangle which includes the machining and CBV areas. The volume will be calculated using Divergence(Gaussian) theorem by focusing on the direction of the normal vector on each triangle. This method can be used as an alternative to selecting tothe rough machining methods which select minimum value of nonmachinable volume so that effectiveness can be achieved in the machining process.