Imposed Boundary Conditions Research Articles

Enhanced physics-informed neural networks without labeled data for weakly and fully coupled transient thermomechanical analysis

In this paper, a series of enhanced physics-informed neural networks (PINN) models without labeled data is proposed to solve the weakly and fully coupled thermomechanical problems. In these models, to better predict the thermal and mechanical responses, PINNs consisting of different deep neural networks (DNN) representing temperature, displacement, and stress are specifically constructed. Furthermore, to elevate the accuracy and avoid possible training failure, several advanced algorithms are developed to ensure the effectiveness of imposing boundary conditions, refining sampling distributions, and enhancing training strategy. A notable aspect of the enhanced PINNs is their independence from expensive, labeled data, relying solely on the temporal and spatial information embedded within the sampling points. The effectiveness and accuracy of the enhanced PINNs are validated through extensive numerical examples, including heat conduction and both weakly and fully coupled thermomechanical problems. The comparation between original PINN and enhanced PINN illustrates the necessity of involving these enhanced methods. The results demonstrate the significant potential of PINN methodologies in engineering areas involving complex thermomechanical processes.

DISPERSION OF LAMB WAVES IN MULTILAYER STRUCTURES

Low cost, the possibility of online monitoring and high sensitivity distinguish the method of structural monitoring using Lamb waves from other available methods. Structural analysis based on Lamb waves in heterogeneous materials requires fundamental knowledge of the behavior of Lamb waves in such materials. This basic knowledge is critical for signal processing in determining possible damage that can be detected by the propagating wave. Recently, Lamb wave methods have been used to simultaneously survey large areas of composite structures. However, such methods are more complex than traditional ultrasonic testing because Lamb waves have dispersive characteristics, namely, the wave speed varies depending on the frequency, modes and thickness of the plates. Experimentally measured group velocities of Lamb waves in composite materials with anisotropic characteristics do not coincide with theoretical group velocities, which are calculated using the dispersion equation of Lamb waves. This discrepancy arises because in anisotropic materials there is an angle between the direction of the group velocity and the direction of the phase velocity. This work investigates the propagation characteristics of Lamb waves in composites, focusing on group velocity and characteristic wave curves. For symmetric laminates, a robust method is proposed by imposing boundary conditions on the mid-plane and top surface to separate symmetric and antisymmetric wave modes. The dispersive and anisotropic behavior of Lamb waves in two different types of symmetrical laminates is theoretically studied in detail. The dispersion of Lamb waves was studied for 10 symmetric and asymmetric modes. It is shown that only fundamental modes are not characterized by a cutoff frequency, which indicates the interaction of fundamental modes with composite layers in the low-frequency range. A high level of group velocity dispersion was discovered for the SH0 and S0 modes. It is concluded that in isotropic laminates, dispersion is characteristic of symmetric modes. It is shown that the frequency dependence of the group velocity of Lamb waves of laminar composites can be represented in polynomial form.

Total and symmetry resolved entanglement spectra in some fermionic CFTs from the BCFT approach

In this work, we study the universal total and symmetry-resolved entanglement spectra for a single interval of some 2d Fermionic CFTs using the Boundary Conformal Field theory (BCFT) approach. In this approach, the partition of Hilbert space is achieved by cutting out discs around the entangling boundary points and imposing boundary conditions preserving the extended symmetry under scrutiny. The reduced density moments are then related to the BCFT partition functions and are also found to be diagonal in the symmetry charge sectors. In particular, we first study the entanglement spectra of massless Dirac fermion and modular invariant Z2-gauged Dirac fermion by considering the boundary conditions preserving either the axial or the vector U(1) symmetry. The total entanglement spectra of the modular invariant Z2-gauged Dirac fermion are shown to match with the compact boson result at the compactification radius where the Bose-Fermi duality holds, while for the massless Dirac fermion, it is found that the boundary entropy term doesn’t match with the self-dual compact boson. The symmetry-resolved entanglement is found to be the same in all cases, except for the charge spectrum which is dependent on both the symmetry and the theory. We also study the entanglement spectra of N massless Dirac fermions by considering boundary conditions preserving different chiral U(1)N symmetries. Entanglement spectra are studied for U(1)M subgroups, where M ≤ N, by imposing boundary conditions preserving different chiral symmetries. The total entanglement spectra are found to be sensitive to the representations of the U(1)M symmetry in the boundary theory among other behaviours at O(1). Similar results are also found for the Symmetry resolved entanglement entropies. The characteristic log log (ℓ/ϵ) term of the U(1) symmetry is found to be proportional to M in the symmetry-resolved entanglement spectra.

A hybrid interpolating element-free Galerkin method for 3D steady-state convection diffusion problems

This study investigates a hybrid interpolating element-free Galerkin (HIEFG) method for solving 3D convection diffusion problems. The HIEFG approach divides a 3D solution domain into a sequence of interconnected 2D sub-domains, and in these 2D sub-domains, the interpolating element-free Galerkin (IEFG) method is applied to form the discretized equations. The improved interpolating moving least-squares (IMLS) method is used to obtain the shape function of the IEFG method for 2D problems. The finite difference method is employed to combine the discretized equations in 2D sub-domains in the splitting direction. Then, the HIEFG method's formulas are derived for steady-state convection diffusion problems in 3D solution domain. Three numerical examples are used to discuss the impacts of the number of nodes, the number of split layers, and the scaling parameters of the influence domain on the computational precision and CPU time of the HIEFG technique. Imposing boundary conditions directly and the dimension splitting technique in this method significantly improves the computational speed greatly.

Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes

Abstract It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.

Stable neural network-based traveltime tomography using hard-constrained measurements

Traveltime tomography, or travel time inversion, has been one of the primary seismological tools for decades and has been used to understand Earth's properties and dynamic processes. An accurate, preferably flexible, eikonal solver to compute the travel time field is at the heart of the inversion process. However, most conventional eikonal solvers suffer from first-order convergence errors and difficulties dealing with irregular computational grids. Physics-informed neural networks (PINNs) have been introduced to tackle these problems and have successfully addressed those challenges. Nevertheless, these approaches still suffer from slow convergence and unstable training dynamics due to the multi-term nature of the loss function. To improve this, we propose a new formulation for the isotropic eikonal equation, which imposes boundary conditions as hard constraints. We employ the theory of functional connections to the traveltime tomography problem, which allows for the use of a single loss term for training the PINN model. Its efficiency, stability, and flexibility in tackling various cases, including topography-dependent and 3D models, are attested through rigorous numerical tests, thus providing an efficient and stable PINN-based traveltime tomography.

Upwind Summation-by-parts Finite Differences: error Estimates and WENO methodology

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous approximation term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

Geometry-independent spline finite element method to analyze two-dimensional heat conduction and elasticity problems

Traditional isogeometric analysis involves redundant geometric calculations and is more complex than the finite element method when imposing boundary conditions. To address this, a geometry-independent spline finite element method is proposed, which integrates the geometric construction techniques of isogeometric analysis and the element construction techniques of finite element method. Firstly, the geometrical precision element is constructed by using non-uniform rational B-spline and B-splines to describe the geometry and the physical field, where a transformation matrix is introduced to construct shape functions with interpolation properties. Secondly, calculation formats for two-dimensional heat conduction and elasticity problems are derived by parameter mapping. Finally, the feasibility and accuracy of the method are verified through several numerical examples. Unlike in isogeometric analysis, in the proposed method, geometry and physics are separated, and the shape functions have interpolation properties, which reduce redundant geometry calculations and allow the boundary conditions to be applied directly to the nodes. The numerical results indicate that the method has a higher convergence rate compared to the original isogeometric analysis.

Two‐dimensional controlled source electromagnetic inversion algorithm based on a space domain forward modeling approach

AbstractWe develop a two‐dimensional controlled‐source electromagnetic inversion algorithm employing a space domain forward modelling algorithm. The space domain forward modelling algorithm is devised by imposing boundary conditions on the plane perpendicular to the strike direction that passes through the source position. The boundary conditions for various source types are derived using the symmetric/antisymmetric character of the electric and magnetic fields. The benchmarking analysis reveals that roughly eight grids are sufficient for discretizing space in the strike directions for accurate forward response computations. For inverse modelling, the inexact Gauss–Newton optimization technique is utilized. Numerical inversion experiments of synthetic and real‐field data clearly demonstrate the versatility and robustness of the developed algorithm. The inversion experimentations also concur with the forward response benchmarking analysis and suggest that only a few grids (around eight) are adequate to discretize space in the strike direction. The developed algorithm is more than one order efficient compared to a wavenumber domain code.

Finite difference delay modelling for analysis of shielding effectiveness of perforated conductive enclosure

AbstractThis paper introduces finite difference delay modelling (FDDM) for computing the shielding effectiveness (SE) of a conductive enclosure with an aperture against electromagnetic pulses. FDDM offers optimal accuracy and stability for analysing complex structures. The time domain electric field integral equation (TD‐EFIE) for the conductive enclosure is derived by imposing boundary conditions on the perfect electrical conductor (PEC) surface in the Laplace domain. Time discretization is based on finite differences, and the Laplace‐to‐Z transform mapping is utilized. Fast Fourier Transform (FFT) is employed to expedite the FDDM solution process. Both frequency domain shielding effectiveness (FD‐SE) and time domain shielding effectiveness (TD‐SE) are evaluated using this approach. Finally, to validate the accuracy of the proposed method, results are compared with simulations using CST‐MWS software and the frequency domain moment method.

A multiscale stabilized physics informed neural networks with weakly imposed boundary conditions transfer learning method for modeling advection dominated flow

A multiscale stabilized physics informed neural networks with weakly imposed boundary conditions transfer learning method for modeling advection dominated flow

Mixed boundary conditions and double-trace like deformations in Celestial holography and Wedge-like holography

According to the AdS/CFT dictionary, adding a relevant double-trace deformation f ∫ O2 to a holographic CFT action is dual to imposing mixed Neumann/Dirichlet boundary conditions for the field dual to O in AdS. We observed similar behaviour in codimension-two flat space holographies. We consider deformations of boundary conditions in flat spacetimes under flat space codimension-two holographies, Celestial holography and Wedge-like holography. In the former Celestial-holographic approach, we imposed boundary conditions on initial and final bulk states in the scattering. We find that these non-trivial boundary conditions in the bulk induce “double deformations” on the Celestial CFT side, which can be understood as an analogy of double trace deformations in the usual AdS/CFT. We compute two-point bulk scattering amplitudes under the non-trivial deformed boundary conditions. In the latter Wedge-like holography approach, we consider mixed Neumann/Dirichlet boundary conditions on the null infinity of the light-cone. We find that this mixing induces a renormalization flow in the dual Wedge CFT side under the Wedge holography, as in the usual AdS/CFT. We argue that the discrepancy between the Wedge two-point function and the Celestial two-point function originates from a sensitivity of bulk massless fields to a regularization parameter to use the usual AdS/CFT techniques.

Cosmic-Ray Acceleration of Galactic Outflows in Multiphase Gas

We investigate the dynamical interaction between cosmic rays (CRs) and the multiphase interstellar medium (ISM) using numerical magnetohydrodynamic (MHD) simulations with a two-moment CR solver and TIGRESS simulations of star-forming galactic disks. We previously studied the transport of CRs within TIGRESS outputs using a “postprocessing” approach, and we now assess the effects of the MHD backreaction to CR pressure. We confirm our previous conclusion that there are three quite different regimes of CR transport in multiphase ISM gas, while also finding that simulations with “live MHD” predict a smoother CR pressure distribution. The CR pressure near the midplane is comparable to other pressure components in the gas, but the scale height of CRs is far larger. Next, with a goal of understanding the role of CRs in driving galactic outflows, we conduct a set of controlled simulations of the extraplanar region above z = 500 pc, with imposed boundary conditions flowing from the midplane into this region. We explore a range of thermal and kinematic properties for the injected thermal gas, encompassing both hot, fast-moving outflows, and cooler, slower-moving outflows. The boundary conditions for CR energy density and flux are scaled from the supernova rate in the underlying TIGRESS model. Our simulations reveal that CRs efficiently accelerate extraplanar material if the latter is mostly warm/warm-hot gas, in which CRs stream at the Alfvén speed, and the effective sound speed increases as density decreases. In contrast, CRs have very little effect on fast, hot outflows where the Alfvén speed is small, even when the injected CR momentum flux exceeds the injected MHD momentum flux.

Dynamical actions and q-representation theory for double-scaled SYK

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SUq(1, 1). We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SUq(1, 1), whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the q-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.

Solar Eruptions Triggered by Flux Emergence below or near a Coronal Flux Rope

Observations have shown a clear association of filament/prominence eruptions with the emergence of magnetic flux in or near filament channels. Magnetohydrodynamic (MHD) simulations have been employed to systematically study the conditions under which such eruptions occur. These simulations to date have modeled filament channels as 2D flux ropes or 3D uniformly sheared arcades. Here we present MHD simulations of flux emergence into a more realistic configuration consisting of a bipolar active region containing a line-tied 3D flux rope. We use the coronal flux-rope model of Titov et al. as the initial condition and drive our simulations by imposing boundary conditions extracted from a flux emergence simulation by Leake et al. We identify three mechanisms that determine the evolution of the system: (i) reconnection displacing footpoints of field lines overlying the coronal flux rope, (ii) changes of the ambient field due to the intrusion of new flux at the boundary, and (iii) interaction of the (axial) electric currents in the preexisting and newly emerging flux systems. The relative contributions and effects of these mechanisms depend on the properties of the preexisting and emerging flux systems. Here we focus on the location and orientation of the emerging flux relative to the coronal flux rope. Varying these parameters, we investigate under which conditions an eruption of the latter is triggered.

Boundary driven turbulence on string worldsheet

We study the origin of turbulence on the string worldsheet with boundaries laid in anti de Sitter (AdS) spacetime. While the classical motion of a single closed string in AdS is integrable, it has recently been recognized that weak turbulence arises in the case of an open string suspended from the AdS boundary. In the open string case, it is necessary to impose boundary conditions on the worldsheet boundaries. We classify which boundary conditions preserve integrability. Based on this classification, we anticipate that turbulence may occur on the string worldsheet if integrability is not guaranteed by the boundary conditions. Numerical investigations of the classical open-string dynamics support that turbulence occurs when the boundary conditions are not integrable.

A nonlocal energy-informed neural network for peridynamic correspondence material models

This paper presents a nonlocal energy-informed neural network to address the inherent zero-energy mode instability in peridynamic correspondence material models. Starting from the principle of virtual work, the task of solving peridynamic equilibrium equation is reformulated as a variational minimum problem of total potential energy, serving as the natural loss function for neural network. By minimizing the potential energy of the peridynamic body, the proposed neural network effectively eliminates the zero-energy modes. Importantly, this approach preserves the original peridynamic formulation without introducing a supplementary force vector state or modifying the nonlocal deformation gradient tensor. Additionally, there is no need for an artificial damping coefficient in adaptive dynamic relaxation, and the high memory consumption in implicit schemes is avoided. To improve the prediction accuracy near boundaries and to impose boundary conditions consistently with the local continuum theory, variable horizon method is introduced. Comparative studies of the proposed neural network with analytical solutions, finite element analyses or other control methods are conducted to verify the feasibility and accuracy of the proposed neural network in suppressing zero-energy mode oscillations. Furthermore, the convergence of the proposed approach and the influence of different activation functions on the prediction accuracy and convergence rate are investigated.

An explicit Jacobian for Newton's method applied to nonlinear initial boundary value problems in summation-by-parts form

<p>We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.</p>

Interaction between forced and natural convection in a thin cylindrical fluid layer at low Prandtl number

Motivated by nuclear safety issues, we study the heat transfers in a thin cylindrical fluid layer with imposed fluxes at the bottom and top surfaces (not necessarily equal) and a fixed temperature on the sides. We combine direct numerical simulations and a theoretical approach to derive scaling laws for the mean temperature and for the temperature difference between the top and bottom of the system. We find two asymptotic scaling laws depending on the flux ratio between the upper and lower boundaries. The first one is controlled by heat transfer to the side, for which we recover scaling laws characteristic of natural convection (Batchelor, Q. Appl. Maths, vol. 12, 1954, pp. 209–233). The second one is driven by vertical heat transfers analogous to Rayleigh–Bénard convection (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56). We show that the system is inherently inhomogeneous, and that the heat transfer results from a superposition of both asymptotic regimes. Keeping in mind nuclear safety models, we also derive a one-dimensional model of the radial temperature profile based on a detailed analysis of the flow structure, hence providing a way to relate this profile to the imposed boundary conditions.

A trial solution for imposing boundary conditions of partial differential equations in physics-informed neural networks

This article proposes an auxiliary function for imposing the boundary and initial conditions in physics-informed neural network models in a hard manner that accelerates the learning process. This auxiliary function consists of two pre-trained neural networks and a main deep neural network with trainable parameters. The novelty of this new auxiliary function is the input of main deep neural network, which takes the outputs of distance function and boundary function as inputs in addition to the spatiotemporal variables. We demonstrate the efficacy and general applicability of the proposed model by applying it to several benchmark-forward problems namely, advection, Helmholtz and Klein-Gordon equations. The accuracy of predictions is examined by comparison with exact solutions. Our findings imply the superiority of the proposed model because of the improvement of the loss convergence to lower values by one order of magnitude for the same number of epochs. In the case of the advection equation, the relative L2 error has been reduced from 0.025 to 0.0201, and from 0.016 to 0.0152. When applied to the Helmholtz equation, our novel model achieved an error of 8.67 × 10−3, surpassing the conventional model's performance, which yielded an error of 6.04 × 10−2. Furthermore, for the Klein-Gordon equation, our new model led to a remarkable reduction in the relative L2 error, from 0.18 to an impressive 4.2 × 10−2.