Singular Forces Research Articles

A MAC grid based FFT-AMIB solver for incompressible Stokes flows with interfaces and singular forces

In this paper, a second order accurate Augmented Matched Interface and Boundary (AMIB) method is introduced for solving the incompressible Stokes flows with interfaces and singular forces. Based on the MAC staggered grid for velocity and pressure, the zeroth order and first order jump conditions are repeatedly enforced in the AMIB scheme to generate fictitious values around the interface, which allow an accurate approximation of the Cartesian derivative jumps. By treating all jump quantities as auxiliary variables and coupling them with the corrected finite difference formulas, the Stokes equations can be discretized into an augmented system. The proposed FFT-AMIB solver does not need to take the pressure boundary condition into account. Moreover, in the Schur complement solution of the augmented system, the discrete Laplacian can be efficiently inverted by using the fast Fourier transform (FFT). Numerical results show that the FFT-AMIB solver is very efficient and can achieve second order accuracy for both velocity and pressure. The iteration number of Bi-CGSTAB in solving the resulting augmented system only weakly depends on the mesh size. When the interface is allowed to be moved under the surface tension, the proposed algorithm can maintain the divergence-free condition and area conservation very well.

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.

A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

Thermosolutal Convection of Natural and Anti-Natural Solutions Through an Angled Cavity Under Cross Gradients in Temperature and Concentration

In the current study, cross-temperature and concentration gradients are used to model the in a binary fluid contained in an angled square cavity. Using a method, the , , and conservation equations were numerically solved. The inclined cavity under equal solutal buoyancy and thermal forces was the subject of the study . Since the horizontal components of the thermal and singular volume forces were equal but opposed to one another, an equilibrium solution for this situation that corresponds to the rest state of the immobile fluid is feasible. However, this equilibrium solution becomes unstable above a specific critical value of the , leading to vertical density stratification inside the enclosure. The results are shown using the and as well as the and for the flow intensity. The existence of the commencement of convection is demonstrated in this work, and both natural and anti-natural flow solutions are obtained. Subcritical convection has also been seen for the natural solution when the is more or less than unity. For the start of supercritical and subcritical convection, the number's critical values are identified. As the climbed, so did the flow's intensity and the rates at which heat and mass were transferred. Reducing flow intensity and accelerating mass transfer are the results of raising the . Different flow patterns are shown for an aspect ratio of 4, and the existence interval of the oscillatory solutions is calculated.

Unfitted finite element methods based on correction functions for Stokes flows with singular forces of low regularity

This paper presents an unfitted finite element method based on correction functions for solving stationary Stokes flows with singular forces acting on an immersed interface. It has been shown that the singular force is equivalent to a nonhomogeneous jump condition on the interface. In this paper, we consider the case that the jump has a low regularity so that it is impossible to use pointwise values on the interface to construct correction functions, as done in Guzmán et al. (2016). The natural way to deal with the problem is to use mean values of the jump on the parts of the interface cut by elements, instead of using pointwise values. However, we show that it may cause instability and the constant in the error estimate may depend on the interface location relative to the mesh. Inspired by Guo et al. (2019), we use a larger fictitious circle to overcome these issues. Associated with the correction functions, we consider two stable finite element pairs: the Mini element and the P2−P0 element, including the cases of continuous and discontinuous pressures. The optimal approximation capabilities of the correction functions and optimal error estimates of the finite element methods are both derived with a hidden constant independent of the interface location relative to the mesh. Numerical examples are provided to validate the theoretical results.

An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.

Singular Points and Singular Curves in von Kármán Elastic Surfaces

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated defects (e.g., dislocations and disclinations), and singular metric anomaly fields (e.g., growth and thermal strains). With such concerns as our motivation, we model thin elastic surfaces as von Kármán plates and generalize the classical von Kármán equations, which are restricted to smooth fields, to fields which are piecewise smooth, and can possibly concentrate at singular curves, in addition to being singular at isolated points. The inhomogeneous sources to the von Kármán equations, given in terms of plastic strains, defect induced incompatibility, and body forces, are likewise allowed to be singular at isolated points and curves in the domain. The generalized framework is used to discuss the singular nature of deformation and stress arising due to conical deformations, folds, and folds terminating at a singular point.

Quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with singular interaction forces

We establish a quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean–Vlasov equation. Introducing an intermediate system via a coarse-graining map, we quantitatively estimate the error between the spatial densities of the Vlasov–Fokker–Planck equation and the intermediate system in the Wasserstein distance of order 2. We then derive an evolution-variational-like inequality for Wasserstein gradient flows which allows us to quantify the error between the intermediate system and the corresponding limiting equation. Our strategy only requires weak integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov–Poisson–Fokker–Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.

Stability of singular solutions to the Navier-Stokes system

We develop mathematical methods which allow us to study asymptotic properties of solutions to the three dimensional Navier-Stokes system for incompressible fluid in the whole three dimensional space. We deal either with the Cauchy problem or with the stationary problem where solutions may be singular due to singular external forces which are either singular finite measures or more general tempered distributions with bounded Fourier transforms. We present results on asymptotic properties of such solutions either for large values of the space variables (so called the far-field asymptotics) or for large values of time.

Quantitative estimate of the overdamped limit for the Vlasov–Fokker–Planck systems

This note adapts a probabilistic approach to establish a quantified estimate of the overdamped limit for the Vlasov–Fokker–Planck equation towards the aggregation–diffusion equation, which in particular includes cases of the Newtonian type singular forces. The proofs are based on the investigation of the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space. We show that one can recover the same (actually stronger) overdamped limit result as in Choi and Tse (2020) under the same assumptions.

A simple augmented IIM for 3D incompressible two-phase Stokes flows with interfaces and singular forces

In this paper, a Stokes solver on a MAC staggered grid for three-dimensional incompressible two-phase flows with interfaces and singular forces is presented. The main purpose is to develop the simplified IIM to ensure the accuracy near the interface represented by a level-set function. The idea of the simplified IIM is to apply Taylor expansion only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference scheme. A Newton iteration method is applied to approximate the orthogonal projections for points near the interface. The augmented variables are introduced to decouple the coupled jump conditions of velocity and pressure. The determination of the augmented variables is finally reduced to a linear system, which is solved by the GMRES method. A CG-Uzawa type method is used to solve the discetized Stokes equation with some correction terms. Numerical results show that the proposed algorithm is efficient and can reach global second-order accuracy. The present algorithm can also preserve the area conservation and the discrete divergence free condition very well.

Star flows and multisingular hyperbolicity

A vector field X is called a star flow if every periodic orbit of any vector field C^1 -close to X is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3- or 4-manifold are either hyperbolic, or singular hyperbolic (see [MPP] for 3-manifolds and [LGW] for 4-manifolds). As it is defined, the notion of singular hyperbolicity forces the singularities in the same class to have the same index. However in higher dimensions (i.e. ≥ 5 ), [dL1] shows that singularities of different indices may be robustly in the same chain recurrence class of a star flow. Therefore the usual notion of singular hyperbolicity is not enough for characterizing the star flows. We present a form of hyperbolicity (called multisingular hyperbolicity ) which makes the hyperbolic structure of regular orbits compatible with the one of singularities even if they have different indices. We show that multisingular hyperbolicity implies that the flow is star, and conversely we prove that there is a C^1 -open and dense subset of the open set of star flows which are multisingular hyperbolic. More generally, for most of the hyperbolic structures (dominated splitting, partial hyperbolicity etc.) well defined on regular orbits, we propose a way of generalizing it to a compact set containing singular points.

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

Computing viscous flow along a 2D open channel using the immersed interface method

AbstractWe present a numerical method for simulating 2D flow through a channel with deformable walls. The fluid is assumed to be incompressible and viscous. We consider the highly viscous regime, where fluid dynamics are described by the Stokes equations, and the less viscous regime described by the Navier–Stokes equations. The model is formulated as an immersed boundary problem, with the channel defined by compliant walls that are immersed in a larger computational fluid domain. The channel traverses through the computational domain, and the walls do not form a closed region. When the walls deviate from their equilibrium position, they exert singular forces on the underlying fluid. We compute the numerical solution to the model equations using the immersed interface method, which preserves sharp jumps in the solution and its derivatives. The immersed interface method typically requires a closed immersed interface, a condition that is not met by the present configuration. Thus, a contribution of the present work is the extension of the immersed interface method to immersed boundary problems with open interfaces. Numerical results indicate that this new method converges with second‐order accuracy in both space and time, and can sharply capture discontinuities in the fluid solution.

An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains.

In this paper, we present a deep learning framework for solving two-dimensional elliptic equations with singular forces on arbitrary domains. This work follows the ideas of the physical-inform neural networks to approximate the solutions and the immersed boundary method to deal with the singularity on an interface. Numerical simulations of elliptic equations with regular solutions are initially analyzed in order to deeply investigate the performance of such methods on rectangular and irregular domains. We study the deep neural network solutions for different number of training and collocation points as well as different neural network architectures. The accuracy is also compared with standard schemes based on finite differences. In the case of singular forces, the analytical solution is continuous but the normal derivative on the interface has a discontinuity. This discontinuity is incorporated into the equations as a source term with a delta function which is approximated using a Peskin's approach. The performance of the proposed method is analyzed for different interface shapes and domains. Results demonstrate that the immersed boundary neural network can approximate accurately the analytical solution for elliptic problems with and without singularity.

Laminar flow past an infinite planar array of fixed particles: point-particle approximation, Oseen equations and resolved simulations

In the point-particle model of disperse multiphase flow, the particles, assumed to be very small compared with all the scales of the flow, are represented by singular forces acting on the fluid. The hydrodynamic forces are found from standard correlations by interpolating the velocity field from the grid nodes to the particle positions, with the implicit assumption that the computational cells are much larger than the particles. It is argued here that this model has similarities with the Oseen linearization of the Navier–Stokes equation, the most important one being that, in the Oseen context, the particles are also treated, to leading order, as singularities. For this and other reasons addressed in the paper, the Oseen equations can be used as proxies for the point-particle model and the comparison of their solutions with particle-resolved simulations, both of which are presented in this paper, can shed light on the strengths and weaknesses of the point-particle model. The specific situation considered is the laminar, steady, uniform flow normal to a plane of periodically arranged particles exchanging momentum and heat with the fluid.

Numerical investigation of the rheological behavior of a dense particle suspension in a biviscous matrix using a lubrication dynamics method

This paper presents a numerical approach to predict the rheology of dense non-colloidal suspensions with a biviscous matrix. A biviscous matrix is characterized as a fluid with two shear rate dependent viscosities i.e. one above and below a critical shear rate γ˙c. The methodology is based on the lubrication dynamics which dominantly influence the suspension properties at high values of particle concentration. To efficiently handle the singular lubrication forces in the dense suspensions, a semi-implicit splitting integration scheme is employed. Using the presented approach, three dimensional simulations were performed and the predicted rheology of the suspension with a biviscous matrix is discussed under two regimes: (a) γ˙c larger than the macroscopic applied shear rate where fluid slippage effect can be modeled in terms of the non-Newtonian properties of the matrix, and (2) γ˙c smaller than the macroscopic applied shear rate where a biviscous model can be seen as a regularization of an apparent yield stress matrix. The results obtained at high γ˙c show that the shear thinning of the biviscous matrix in the inter-particle gaps, which can be interpreted as an apparent fluid slipping on the particle surface, provides an alternative mechanism to explain the experimentally observed shear-thinning of non-colloidal suspension with Newtonian matrices. At low γ˙c values, the predicted suspension properties and its microstructure corroborates the available experimental results on suspensions with yield stress fluids.

Emergence of the Consensus and Separation in an Agent-Based Model With Attractive and Singular Repulsive Forces

In this paper, we study an agent-based interacting particle system with attractive and singular repulsive forces. We prove the collision avoidance between particles from different groups due to rep...

Finite element method for incompressible viscous flow with immersed pressure jumps with applications to actuator disks and microfluidics

We propose a finite element method for the solution of viscous incompressible flow problems with singular forces at immersed interfaces. The method combines the algebraic subgrid scale method with a pressure jump stabilization. It consists of the addition, to the continuity equation, of a term weighting the residual of the pressure jump. This term enhances the stability irrespective of possible badly shaped intersections of the interface with the finite elements. We assess the new method by comparing with the unstabilized case showing improved accuracy and robustness. The examples consider immersed actuator disk problems and one application to thermocapillary convection.

Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off

We consider a [Formula: see text]-particle system interacting through the Newtonian potential with a polynomial cut-off in the presence of noise in velocity. We rigorously prove the propagation of chaos for this interacting stochastic particle system. Taking the cut-off like [Formula: see text] with [Formula: see text] in the force, we provide a quantitative error estimate between the empirical measure associated to that [Formula: see text]-particle system and the solutions of the [Formula: see text]-dimensional Vlasov–Poisson–Fokker–Planck (VPFP) system. We also study the propagation of chaos for the Vlasov–Fokker–Planck equation with less singular interaction forces than the Newtonian one.