Helmholtz Decomposition Research Articles

Motion Estimation for Complex Fluid Flows Using Helmholtz Decomposition

In this paper, we proposed a novel motion model with Helmholtz decomposition for complex fluid flows in a filtering-based optical flow framework, where the optimization of the regularization term is treated as a filtering process, and different motion patterns can be captured by finding appropriate filter kernels based on the designed filters. In this framework, we introduce a novel optical flow method with a joint spatial filter, which is based on the Helmholtz decomposition theorem that assumes a local motion field is composed of a curl field and a divergence field. By adjusting the scale of the weights in the filter kernels and combining a curl filter with a divergence filter in a certain ratio, it can simulate different motion patterns. In addition, if the correlation between the horizontal and vertical components of the optical flow field in the filter kernel is eliminated, it will be transformed into a linear motion model. Based on this linear motion model, we also develop a novel optical flow method with an adaptive guided filter. By finding an adaptive filter kernel driven by both the input image and the guided motion field for the designed filter, it can successfully capture different motion patterns, and yield an edge-preserving smoothing optical flow field. Most importantly, the proposed optical flow model provides a new way to design the regularizers for capturing different motion patterns in complex fluid flow. In particular, the designed optical flow method with an adaptive guided filter significantly outperforms the current state-of-the-art optical flow methods in predicting complex fluid flows.

Compatible finite element methods for geophysical fluid dynamics

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

Effect of flow–thermodynamics interactions on the stability of compressible boundary layers: insights from Helmholtz decomposition

Helmholtz decomposition of velocity perturbations is performed in conjunction with linear stability analysis to examine the effects of flow-thermodynamics interactions on the stability of high-speed boundary layers. A corresponding decomposition of the pressure field is also proposed. The contributions of perturbation solenoidal kinetic, dilatational kinetic and internal energy to the various instability modes are examined as a function of Mach number ( $M$ ). As expected, dilatational and pressure field effects play an insignificant part in the first-mode behaviour at all Mach numbers. The second (Mack) mode, however, is dominantly dilatational in nature, and perturbation internal energy is significant compared to perturbation kinetic energy. The observed behaviour is explicated by examining the key processes of production and pressure-dilatation. Production of the second-mode dilatational kinetic energy is mostly due to the solenoidal-dilatational covariance stress tensor interacting with the mean (background) velocity gradient. This cross production component also inhibits the first mode. The dilatational pressure facilitates energy transfer from the kinetic to the internal field in the near‐wall region, whereas the energy transfer away from the wall is mostly due to the solenoidal pressure work. The dilatational characters of the fast and slow modes are also examined. The fast mode is dominantly dilatational at both $M=4$ and $M=6$ , while the nature of the slow mode is dependent on $M$ . Finally, Helmholtz decomposition of the perturbation momentum vector is performed. Interestingly, both first and second modes are dominated by solenoidal components of momentum.

The Helmholtz Decomposition of a BMO Type Vector Field in a Slightly Perturbed Half Space

We introduce a space of L^2 vector fields with bounded mean oscillation whose “normal” component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed C^3 half space in {textbf{R}}^n(n ge 3) with small perturbation.

A Discontinuous Petrov–Galerkin Method for Reissner–Mindlin Plates

We present a discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.

P/S-wave separation of multicomponent seismic data at the land surface based on deep learning

P/S-wave separation is a key step for data processing in multicomponent seismic exploration. The conventional methods rely on either the prior information of near-surface elastic properties or the carefully selected parameters to estimate the polarization directions of the P- and S-modes when arriving at the geophones. In the case of complex wave propagation or the near-surface elastic parameters being unavailable, the conventional methods fail to achieve a high-quality P/S separation. The P/S-wave separation of multicomponent seismic data recorded at the land surface can be regarded as a nonlinear problem of point-by-point image prediction and addressed by using the powerful ability for feature extraction of a deep neural network. To obtain an accurate, abundant, and representative labeled data set, a variety of elastic models are constructed through geologically reasonable data augmentation, followed by elastic forward modeling and Helmholtz decomposition. A convolutional neural network is trained with these labeled data to predict scalar P- and S-wave recordings from the multicomponent data in a data-driven manner. Eventually, a novel approach of P/S-wave separation is established for multicomponent seismic imaging with the help of deep learning, which can work without prior knowledge of the near-surface elastic parameters. Numerical examples indicate that, by constructing an augmented labeled data set for training, our method significantly improves the generalization of the deep neural network and thus enables the trained neural network to achieve more accurate P/S-wave separation than the conventional model-driven method for the target data.

Analytical simulation of the elastic moduli dispersion for an isotropic porous cylinder

There are extensive laboratory measurements of elastic moduli dispersion of porous materials, yet the mechanisms for the dispersion are not fully understood. Related analytical solutions are missing in the literature, potentially due to complex mathematical derivations. In this paper, we use Biot's theory of poroelastodynamics (PED) and present the first analytical solution for an isotropic fluid-saturated porous cylinder subject to a forced deformation test. This is done by introducing Helmholtz decomposition, two scalar potentials, and two vector potentials to decouple the original governing partial differential equations. The methods of matrix diagonalization and separation of variables are then used to solve the decoupled equations. The elastodynamics (ED) solution is also presented. We demonstrate the coupled responses of displacement, pore pressure, and stress and interpret systematically the mechanisms for the elastic moduli dispersion. Furthermore, we investigate comprehensively the effects of loading frequency, material's poromechanical characteristics, sample size, and boundary conditions on the elastic moduli dispersion. Finally, excellent matches are found between the analytical solution and published measurements of the dynamic Young's moduli and Poisson's ratios of a water-saturated clastic sediment rock and a glycerin-saturated limestone. The analytical solution captures the materials’ elastic moduli dispersion at both seismic and sonic frequencies. This work should be of interest to scientists and engineers who work on dynamics, vibration, and poromechanics.

Modified pressure and vorticity variables using Helmholtz decomposition for solution of the incompressible flow equations

Incompressible Navier–Stokes equations are reformulated using the Helmholtz decomposition of a velocity vector into rotational and potential components. By substituting the decomposed velocity in the time derivative term in a momentum equation, the potential component representing a gradient of a pressure-like term is combined with the gradient of the pressure modifying the physical pressure field. The rotational component representing a curl of a vortex-like vector is combined with the vorticity vector, making a non-physical vorticity vector that modifies the fluid viscosity. Thus, the unsteady Navier–Stokes equation is transformed into an explicitly steady-state form in terms of non-primitive variables. The stream function vector is governed by a parabolic equation in time, while the vorticity vector is governed by the Poisson function with a source term function of the convection stretching and time dependency of the physical flow vorticity. Therefore, the resulting system of equations is numerically independent of the cell Reynolds number stability condition that hunted the convection–diffusion Navier–Stokes equation. Numerical results are obtained for the two-dimensional driven cavity problem for Reynolds number of 1000 with 21 × 21, 41 × 41, 81 × 81, and 161 × 161 grid points. The computational grids correspond to cell Reynolds numbers 25, 12.5, 6.25, and 3.125, respectively. The computed results are smooth in all cases and validate the method.

Helmholtz decomposition and potential functions for n-dimensional analytic vector fields

The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in Rn, we replace the vector potential in R3 by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n−1)/2 rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an n-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka–Volterra equations with n species.

Analysis of fully discrete mixed finite element scheme for stochastic Navier–Stokes equations with multiplicative noise

This paper is concerned with stochastic incompressible Navier–Stokes equations with multiplicative noise in two dimensions with respect to periodic boundary conditions. Based on the Helmholtz decomposition of the multiplicative noise, semi-discrete and fully discrete time-stepping algorithms are proposed. The convergence rates for mixed finite element methods based time-space approximation with respect to convergence in probability for the velocity and the pressure are obtained. Furthermore, with establishing some stability and using the negative norm technique, the partial expectations of the $$H^1$$ and $$L^2$$ norms of the velocity error are proved to converge optimally.

Явное представление сокращенных в размерности уравнений Эйлера сжимаемой жидкости и полной системы уравнений гидродинамики в интегральной форме

Various ways of reducing the complete system of hydrodynamic equations in volume to a system of equations on the surface are of great scientific interest. In this article, the unsteady Euler equations of a compressible fluid and the complete system of hydrodynamic equations are reduced to “stationary” integral-differential equations where time derivatives are absent. Such a procedure (3D → 2D, 2D → 1D) makes it possible to reduce the dimension of the problem by one, which significantly reduces the necessary computing power when modeling them. If to set the correct problem, then one can determine the entire unsteady flow in the volume without solving the unsteady problem. It is enough to set time-varying data only on some surface of this stream, which must be determined independently. The peculiarity of this work is that all equations reduced in dimension are obtained in explicit form, unlike previous works of the authors, where up to 200–500 equations with reduced dimension were proposed, which are very difficult to study and model. In previous authors’ work, dimensionality was reduced by reducing overdetermined systems of differential equations. In this method, in the case of a successful choice of an additional constraint equation for overdetermined systems of differential equations, there is a reduction to PDE systems of dimension less than that of the original PDE systems. The method that is used to reduce the dimension of the Euler equations and the complete system of hydrodynamic equations in this paper is actually a special case of the above. The technique of dimensional reduction in overdetermined systems of PDEs is obviously generalized to the case of integral-differential equations. The article also obtained integral equations in new variables (Lagrangian and pseudo-Lagrangian), which determine the evolution of the flow. Also, a new method is proposed for overdetermination of any system of PDEs using the general integral space relation, which is consequence of the Helmholtz decomposition theorem.

Mixed finite element method for a second order Dirichlet boundary control problem

The main aim of this article is to analyze mixed finite element method for the second order Dirichlet boundary control problem. We develop both a priori and a posteriori error analysis using the energy space based approach. We obtain optimal order a priori error estimates in the energy norm and L2-norm with the help of auxiliary problems. The reliability and the efficiency of proposed a posteriori error estimator is discussed using the Helmholtz decomposition. Numerical experiments are presented to confirm the theoretical findings.

Residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems

AbstractIn this paper, we derive a residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems. The present approach does not involve the Helmholtz decomposition while analyzing the reliability of the estimator. The constants involved in the estimators are independent of the jump of the diffusion coefficient across the interface, and the quasi‐monotonocity assumption on the diffusion coefficient is relaxed. The reliability bound of the estimator consists of the element residual, the edge flux jump and the edge solution jump. The efficiency of the estimator is analyzed by employing a coarsening strategy introduced by Chen and Feng's study. We derive both global upper bound and a local lower bound for the error and an adaptive space–time algorithm is prescribed using the derived estimators. Numerical results illustrating the behavior of the estimators are provided.

Wavefield Separation Algorithm of Helmholtz Theory-Based Discontinuous Galerkin Method Using Unstructured Meshes on GPU

In the field of geophysics, the Helmholtz decomposition (HD) formula is mainly numerically discretized by the finite difference method (FDM), which limits its application to a uniform regular grid only. Few scholars note wavefield separation on unstructured grids. In this study, we aim to develop a wavefield separation algorithm to separate P- and S-wavefields on nonuniform grids. Our scheme is based on an isotropic elastic wave equation. We first transform the HD formula into a weak integral form using the discontinuous Galerkin method (DGM). Then, we consider two types of unstructured meshes—triangle and quadrangle, which are more suitable for complex structures. Moreover, to reduce time costs, the single graphic processor unit (GPU) device is used to improve the computational efficiency. We perform a unified DGM operator by transforming unstructured triangles and quadrangles into standard reference elements using coordinate transformation. Our proposed HD operator enables us to effectively separate P- and S-wavefields on unstructured meshes. We carry out the wavefield separation simulation and calculate the numerical solutions in the homogeneous carbonate model, Graben model, and SEG/EAGE model. The homogeneous model verifies the correctness, availability, and superiority of our proposed separated operator, and the other numerical results show excellent performance for P/S-wavefield separation on unstructured meshes.

A Forward Energy Flux at Submesoscales Driven by Frontogenesis

Abstract Submesoscale currents, comprising fronts and mixed-layer eddies, exhibit a dual cascade of kinetic energy: a forward cascade to dissipation scales at fronts and an inverse cascade from mixed-layer eddies to mesoscale eddies. Within a coarse-graining framework using both spatial and temporal filters, we show that this dual cascade can be captured in simple mathematical form obtained by writing the cross-scale energy flux in the local principal strain coordinate system, wherein the flux reduces to the sum of two terms, one proportional to the convergence and the other proportional to the strain. The strain term is found to cause the inverse energy flux to larger scales while an approximate equipartition of the convergent and strain terms captures the forward energy flux, demonstrated through model-based analysis and asymptotic theory. A consequence of this equipartition is that the frontal forward energy flux is simply proportional to the frontal convergence. In a recent study, it was shown that the Lagrangian rate of change of quantities like the divergence, vorticity, and horizontal buoyancy gradient are proportional to convergence at fronts, implying that horizontal convergence drives frontogenesis. We show that these two results imply that the primary mechanism for the forward energy flux at fronts is frontogenesis. We also analyze the energy flux through a Helmholtz decomposition and show that the rotational components are primarily responsible for the inverse cascade while a mix of the divergent and rotational components cause the forward cascade, consistent with our asymptotic analysis based on the principal strain framework.

Generalized Helmholtz Decomposition for Modal Analysis of Electromagnetic Problems in Inhomogeneous Media

Generalized Helmholtz Decomposition for Modal Analysis of Electromagnetic Problems in Inhomogeneous Media

A boundary integral equation method for the fluid-solid interaction problem

<abstract><p>In this paper, a boundary integral equation method is proposed for the fluid-solid interaction scattering problem, and a high-precision numerical method is developed. More specifically, by introducing the Helmholtz decomposition, the corresponding problem is transformed into a coupled boundary value problem for the Helmholtz equation. Based on the integral equation method, the coupled value problem is reduced to a system of three coupled hypersingular integral equations. Semi-discrete and fully-discrete collocation methods are proposed for the singular integral equations. The presented method is based on trigonometric interpolation and discretized singular operators applied to differentiated interpolation. The convergence of the method is verified by a numerical experiment.</p></abstract>

A Complete Helmholtz Decomposition on Multiply Connected Subdivision Surfaces and Its Application to Integral Equations

The analysis of electromagnetic scattering in the isogeometric analysis (IGA) framework based on Loop subdivision has long been restricted to simply-connected geometries. The inability to analyze multiply-connected objects is a glaring omission. In this paper, we address this challenge. IGA provides seamless integration between the geometry and analysis by using the same basis set to represent both. In particular, IGA methods using subdivision basis sets exploit the fact that the basis functions used for surface description are smooth (with continuous second derivatives) almost everywhere. On simply-connected surfaces, this permits the definition of basis sets that are divergence-free and curl-free. What is missing from this suite is a basis set that is both divergence-free and curl-free, a necessary ingredient for a complete Helmholtz decomposition of currents on multiply-connected structures. In this paper, we effect this missing ingredient numerically using random polynomial vector fields. We show that this basis set is analytically divergence-free and curl-free. Furthermore, we show that these basis recovers curl-free, divergence-free, and curl-free and divergence-free fields. Finally, we use this basis set to discretize a well-conditioned integral equation for analyzing perfectly conducting objects and demonstrate excellent agreement with other methods.

Hierarchical wave-mode separation in the poroelastic medium using eigenform analysis

In the elastic medium, the scalar and vector P- and S-waves decomposition has been extensively studied and some strategies can be extended to the poroelastic medium to extract P- and S-wavefields. However, there are three propagation modes in the poroelastic medium in Biot’s theory, namely, a fast P wave, a slow P wave, and an S wave. Because the propagation characteristic of a slow P wave is different from that of a fast P wave and S wave, the wavefield separation methods in the elastic medium cannot be directly applied to the poroelastic medium to produce a complete wave-mode separation. Based on the eigenform analysis, we have developed a hierarchical wavefield decomposition method to completely separate S waves and fast and slow P waves in the poroelastic medium. Using the Helmholtz decomposition, we first compute scalar and vector potential wavefields to separate P and S waves. Then, a cross-product operator is proposed to decompose fast and slow P waves based on their different polarization directions. To produce correct amplitudes and phases, we apply another cross-product operator and an amplitude correction term to the separated wavefields. Three numerical examples demonstrate that our method can produce accurate fast P-wave, slow P-wave, and S-wave separation results, and the decomposed fast and slow P waves have the same phases and amplitudes as the P-wave potential wavefields.

Effect of heat source on kinetic energy transfer in compressible homogeneous shear turbulence

The effects of heat sources on kinetic energy transfer in compressible homogeneous shear turbulence are studied using numerical simulations at turbulent Mach numbers 0.1 and 0.4 for two levels of heat source. It is found that the strong heat source can significantly enhance both positive and negative components of subgrid-scale (SGS) kinetic energy flux and pressure–dilatation. After adding a strong heat source, compression motions enhance the positive SGS flux, and expansion motions enhance the negative SGS flux at a low turbulent Mach number. According to the Helmholtz decomposition, we found that the solenoidal and dilatational components of pressure–dilatation and SGS kinetic energy flux are increased greatly by a strong heat source at a low turbulent Mach number. The solenoidal mode plays a dominant role in the kinetic energy transfer process, but the contribution of the dilatational mode is not negligible. The dilatational component of the production term is increased by a strong heat source at a low turbulent Mach number, providing the main source of kinetic energy to the dilatational mode. The strong heat source also enhances the kinetic energy exchange between solenoidal mode and dilatational mode through nonlinear advection at a low turbulent Mach number. Moreover, the strong heat source enhances pressure anisotropy, redistribution of the kinetic energy of two transverse components, and energy transfer from internal energy to the kinetic energy through pressure–dilatation term. At a high turbulent Mach number, the strong heat source has little impact on the solenoidal and dilatational components of kinetic energy transfer terms.