Lagrangian Boundary Conditions Research Articles

A Lagrangian free-stream boundary condition for weakly compressible smoothed particle hydrodynamics

This paper proposes the first Lagrangian free-stream boundary condition for weakly-compressible smoothed particle hydrodynamics (WCSPH), significantly decreasing the domain-size effect for simulating flows around static or moving geometries and increasing computational efficiency. The boundary condition is implemented based on several numerical techniques, as follows. First, a spatio-temporal identification and an ensuring technique precisely identify surface particles without misjudgment. Second, the far-field corrections of density and velocity precisely target surface particles. Third, the improved in- and outlet boundaries guarantee the smooth inflow profile and free-outflow particles without the issue of the truncated kernel. The test cases of flows over a flat plate and around a circular cylinder have validated the accuracy and computational efficiency of the present method. In order to demonstrate its applicability, the method has also been tested with more complex examples, such as a fluid-structure interaction (FSI) problem and a 3-D flow around a sphere.

Comparison of Machining Simulations of Aerospace Alloy Al6061-T6 Using Lagrangian and Smoothed Particle Hydrodynamics Techniques

This research focuses on the study of the simulation capabilities of the lagrangian (LAG) model and Smoothed Particle Hydrodynamics (SPH) model for the orthogonal dry machining of aluminum alloy Al6061-T6. A three-dimensional finite element model was developed and verified using experimental data from the published literature. The numerical models were developed using lagrangian boundary conditions via finite element modeling in ABAQUS/Explicit 6.14. The cutting simulations were carried out at low and medium cutting speeds. Johnson–Cook material constitutive law and Johnson–Cook damage law were used in both models. The numerical methodologies are compared based on cutting forces, chip morphology, shear angle, chip separation criterion, and chip thickness. The findings of the present work show that the LAG model is good for predictions regarding cutting forces and chip morphology, while the SPH model is good for predictions regarding the shear angle and chip thickness. The difference between results generated by both models mainly occurred due to the friction coefficient. The comparative study shown here offers a guidance approach for various numerical models for appropriate parameter analysis.

A compactness theorem for SO(3) anti-self-dual equation with translation symmetry

Motivated by the Atiyah–Floer conjecture, we consider SO(3) anti-self-dual instantons on the product of the real line and a three-manifold with cylindrical end. We prove a Gromov–Uhlenbeck type compactness theorem, namely, any sequence of such instantons with uniform energy bound has a subsequence converging to a type of singular objects which may have both instanton and holomorphic curve components. In particular, we prove that holomorphic curves that appear in the compactification must satisfy the Lagrangian boundary condition, a claim which has been long believed in the literature. This result is the first step towards constructing a natural bounding cochain proposed by Fukaya for the SO(3) Atiyah–Floer conjecture.

Generalized Gelfand–Yaglom formula for a discretized quantum mechanic system

The Gelfand–Yaglom formula relates the regularized determinant of a differential operator to the solution of an initial value problem. Here, we develop a generalized Gelfand–Yaglom formula for a Hamiltonian system with Lagrangian boundary conditions in the discrete and continuous settings. Later, we analyze the convergence of the discretized Hamilton–Jacobi operator and propose a lattice regularization for the determinant.

Effects of liquid fuel/wall interaction on thermoacoustic instabilities in swirling spray flames

Computational prediction of thermoacoustic instabilities arising in gas turbine and aero-engine combustors still remains a challenge especially if fuel is injected in a liquid spray form. This study shows that, in LES of such a combustor, the treatment of the liquid fuel film created on the walls of the injection system affects the mean flame weakly, but modifies the flame dynamics strongly. The configuration used for this work is the experimental setup SICCA-spray available at EM2C laboratory in Paris. First steady spray flame measurements are used to validate the LES Euler-Lagrange approach. Two modelling strategies for the interaction between the liquid fuel and the injector walls are tested with a negligible impact on the flame shape and structure. In the second part the same comparison is applied to another operating condition where a self-sustained thermo-acoustic limit-cycle is experimentally observed. In that case resonant coupling is achieved with LES, confirming the adequacy of the approach but only when the film layer is taken into account. Indeed, contrarily to the stable configuration, the difference between the two Lagrangian boundary conditions is shown to have a major impact on the feedback mechanism leading to the thermoacoustic oscillation.

Hill-type formula for Hamiltonian system with Lagrangian boundary conditions

In this paper, we build up Hill-type formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the N-reversible symmetry periodic orbits in n-body problem naturally, where N is an anti-symplectic orthogonal matrix with N2=I. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Krein-type trace formula and give nontrivial estimation for the eigenvalue problem. Combined with the Maslov-type index theory, we give some new stability criteria for the N-reversible symmetry periodic solutions of Hamiltonian systems. As an application, we study the linear stability of elliptic relative equilibria in planar 3-body problem.

Error estimates for projection-based dynamic augmented Lagrangian boundary condition enforcement, with application to fluid–structure interaction

In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the [Formula: see text] and [Formula: see text] norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed.

Reduced open Gromov–Witten invariants on K3 surfaces and multiple cover formula

In the paper, we study the wall-crossing phenomenon of reduced open Gromov–Witten invariants on K3 surfaces with rigid special Lagrangian boundary condition. As a corollary, we derived the multiple cover formula for the reduced open Gromov–Witten invariants, which is an open analog of the Gopakumar–Vafa conjecture of genus zero.

Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds

Let $X$ be a compact toric K\"ahler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula which equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of $X$, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono.

Compactness in the adiabatic limit of disk vortices

This paper is the first input towards an open analogue of the quantum Kirwan map. We consider the adiabatic limit of the symplectic vortex equation over the unit disk for a Hamiltonian G-manifold with Lagrangian boundary condition, by blowing up the metric on the disk. We define an appropriate notion of stable solutions in the limit, and prove that any sequence of disk vortices with energy uniformly bounded has a subsequence converging to such a stable object. We also proved several analytical properties of vortices over the upper half plane, which are new type of bubbles appearing in our compactification.

$J$-holomorphic curves with boundary in bounded geometry

The fundamental properties of $J$-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a $J$-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We show these inequalities hold uniformly if the geometry of the target symplectic manifold and Lagrangian boundary condition is appropriately bounded.

Chern–Weil Maslov index and its orbifold analogue

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surfacewith boundary, which consists of symplectic vector bundle onand a Lagrangian subbundle on @� as well as its gen- eralization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold inte- rior singularites, via curvature integral, and find also an analogous topological definition in these cases.

Relatively open Gromov-Witten invariants for symplectic manifolds of lower dimensions

Let $(X,\omega)$ be a compact symplectic manifold, $L$ be a Lagrangian submanifold and $V$ be a codimension 2 symplectic submanifold of $X$, we consider the pseudoholomorphic maps from a Riemann surface with boundary $(\Sigma,\partial\Sigma)$ to the pair $(X,L)$ satisfying Lagrangian boundary conditions and intersecting $V$. In some special cases, for instance, under the semi-positivity condition, we study the stable moduli space of such pseudoholomorphic maps involving the intersection data. If $L\cap V=\emptyset$, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution $\phi$ on $X$ such that $L$ is the fixed point set of $\phi$ and $V$ is $\phi$-anti-invariant, then we define the so-called relatively open invariants for the tuple $(X,\omega,V,\phi)$ if $L$ is orientable and dim$X\le 6$. If $L$ is nonorientable, we define such invariants under the condition that dim$X\le4$ and some additional restrictions on the number of marked points on each boundary component of the domain.

A reverse isoperimetric inequality for J-holomorphic curves

We prove that the length of the boundary of a $J$-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real $J$-holomorphic curve. The infimum over $J$ of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be $2\pi$ for the Lagrangian submanifold $\mathbb R P^n \subset \mathbb C P^n.$ We apply our result to prove compactness of moduli of $J$-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.

Natural Convection from Inclined Square Cylinder Using Novel Flexible Forcing IB-LBM Approach

A novel flexible forcing immersed boundary (IB) – lattice Boltzmann method (LBM) – is introduced in this article to simulate a complex natural convection phenomenon. In the conventional IB-LBM schemes, the explicit computation of force density may not ensure the exact no-slip velocity/no-jump temperature boundary condition. This produces unphysical streamline/isotherm penetration into the solid object. In the proposed formulation, an implicit approach is suggested to calculate the force density. Using a single Lagrangian correction term, the velocity and temperature boundary condition is accurately satisfied. The developed numerical scheme is applied to analyze a thermal flow situation in an annulus, formed by a hot inclined square cylinder and a cold enclosure. Flow pattern and heat transfer rate at different inclination angles (0<θ<45 deg) are investigated for a wide range of Rayleigh numbers (Ra) (103 – 106). The observed results confirm that the local heat transfer rate is influenced by θ and Ra.

The orientability problem in open Gromov–Witten theory

We give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators. A special case of this formula resolves the orientability question for spaces of maps from Riemann surfaces with Lagrangian boundary condition. As a corollary, we show that the local system of orientations on the moduli space of J-holomorphic maps from a bordered Riemann surface to a symplectic manifold is isomorphic to the pull-back of a local system defined on the product of the Lagrangian and its free loop space. As another corollary, we show that certain natural bundles over these moduli spaces have the same local systems of orientations as the moduli spaces themselves (this is a prerequisite for integrating the Euler classes of these bundles). We will apply these conclusions in future papers to construct and compute open Gromov-Witten invariants in a number of settings.

The moduli space of twisted holomorphic maps with Lagrangian boundary condition: Compactness

Let (X,ω) be a compact symplectic manifold and L be a compact Lagrangian submanifold. Suppose that (X,L) has a Hamiltonian S1 action with moment map μ. Take an S1-invariant, ω-compatible almost complex structure, and we consider tuples (C,P,A,φ) where C is a smooth bordered Riemann surface of fixed topological type, P→C is an S1-principal bundle, A is a connection on P and φ is a section of P×S1X satisfying ∂¯Aφ=0,ινFA+μ(φ)=c with boundary condition φ(∂C)⊂P×S1L. Here FA is the curvature of A and ν is a volume form on C and c∈iR is a constant. We compactify the moduli space of isomorphism classes of such objects with energy ≤K, where the energy is defined to be the Yang–Mills–Higgs functional ‖FA‖L22+‖dAφ‖L22+‖μ(φ)−c‖L22. This generalizes the compactness theorem of Mundet–Tian (2009) [17] in the case of closed Riemann surfaces.

Knot contact homology

The conormal lift of a link $K$ in $\R^3$ is a Legendrian submanifold $\Lambda_K$ in the unit cotangent bundle $U^* \R^3$ of $\R^3$ with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of $K$, is defined as the Legendrian homology of $\Lambda_K$, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization $\R \times U^*\R^3$ with Lagrangian boundary condition $\R \times \Lambda_K$. We perform an explicit and complete computation of the Legendrian homology of $\Lambda_K$ for arbitrary links $K$ in terms of a braid presentation of $K$, confirming a conjecture that this invariant agrees with a previously-defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.

Disk counting on toric varieties via tropical curves

In this paper, we define two numbers. One is defined by counting tropical curves with a stop, and the other is the number of holomorphic disks in toric varieties with Lagrangian boundary condition. Both of these curves should satisfy some incidence conditions. We show that these numbers coincide. These numbers can be considered as Gromov-Witten type invariants for holomorphic disks, and they have similarities as well as differences to the counting numbers of closed holomorphic curves. We study several aspects of them.

Nontrivial solutions for asymptotically linear hamiltonian systems with Lagrangian boundary conditions

In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.