Dynamic Partial Differential Equations Research Articles

SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks

Dynamics of many complex systems, from weather and climate to spread of infectious diseases, can be described by partial differential equations (PDEs). Such PDEs involve unknown function(s), partial derivatives, and typically multiple independent variables. The traditional numerical methods for solving PDEs assume that the data are observed on a regular grid. However, in many applications, for example, weather and air pollution monitoring delivered by the arbitrary located weather stations of the National Weather Services, data records are irregularly spaced. Furthermore, in problems involving prediction analytics such as forecasting wildfire smoke plumes, the primary focus may be on a set of irregular locations associated with urban development. In recent years, deep learning (DL) methods and, in particular, graph neural networks (GNNs) have emerged as a new promising tool that can complement traditional PDE solvers in scenarios of the irregular spaced data, contributing to the newest research trend of physics informed machine learning (PIML). However, most existing PIML methods tend to be limited in their ability to describe higher dimensional structural properties exhibited by real world phenomena, especially, ones that live on manifolds. To address this fundamental challenge, we bring the elements of the Hodge theory and, in particular, simplicial convolution defined on the Hodge Laplacian to the emerging nexus of DL and PDEs. In contrast to conventional Laplacian and the associated convolution operation, the simplicial convolution allows us to rigorously describe diffusion across higher order structures and to better approximate the complex underlying topology and geometry of the data. The new approach, Simplicial Neural Networks for Partial Differential Equations (SNN PDE) offers a computationally efficient yet effective solution for time dependent PDEs. Our studies of a broad range of synthetic data and wildfire processes demonstrate that SNN PDE improves upon state of the art baselines in handling unstructured grids and irregular time intervals of complex physical systems and offers competitive forecasting capabilities for weather and air quality forecasting.

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes.

Elastodynamic Green's functions for sandwich panels with aluminum foam core and transversely isotropic face sheets using potential functions method

Green's functions of stress and displacement due to wave propagation in a sandwich panel consisting of transversely isotropic face-sheets and aluminum foam core are developed in the present paper. Dynamic partial differential equations of equilibrium for the considered face-sheets and core are expressed in cylindrical coordinates (r, θ, z), and converted into two separate equations using two unknown potential functions. Then, by writing the potential functions as Fourier series in the circumferential direction and using Hankel transform in the radial direction, an analytical solution to the potential functions in Hankel transformed space is developed. Using the boundary conditions and continuity of the problem as well as the place of application of harmonic forces, Green's functions for displacements and stresses in the frequency domain are derived using Hankel's inverse integral transform. These functions are expressed as complex integrals and calculated numerically. To verify the methodology and results, a comparison study is performed for particular cases showing a high level of the accuracy. Green's functions obtained from this paper will give information about the loading models or materials causing the maximum normal stress, shear stress, vertical and radial displacements. In addition, based on the results, with an increase in the core thickness compared to that of face-sheets in a Sandwich panel, an increment in Green's functions of stress and displacement is observed in both real and imaginary parts.

Mindlin cracked plates modelling and implementation in train-track coupled dynamics

Nowadays, the widely constructed ballastless track in high-speed railways will inevitably suffer from localized damage cracks induced by complicate environmental effects, which would change its designed dynamic performance under train dynamic loads and further affect its service life and durability. To this end, a semi-analytical approach of the in-plane and the out of plane forced vibration for Mindlin plates with a side crack are derived, and is further implemented into the train-track coupled dynamic (TTCD) system which comprehensively considering nonlinear wheel-rail contact and 3D spatial flexibility of the cracked track slab. The dynamic partial differential equation of the cracked slab is derived by Hamilton principle and discretized by Galerkin method. Then the eigenvalues and forced vibrations of the cracked slabs for both in-plane and flexural vibrations are tackled utilizing a set of powerful trigonometric modified by special corner function, which can effectively explain the singularity of stress at the crack tip and the discontinuity of deformation on both sides of the crack. The eigenvalue analysis and mode shapes of the cracked plate mode in free boundary condition are compared with the finite element method to verify its accuracy. Meanwhile, the reliability of the proposed dynamics model is verified by comparing with the responses predicted by the co-simulation technology. Finally, the discrepancies of dynamic responses of the track slab with or without a side crack are analyzed, and the effects of various operation speeds and support stiffness on the dynamic behaviour of both slabs are revealed. Results indicates that the track slab with a side crack has significant distinctions on the slab dynamic signals in both time and frequency domains, and the amplitude amplification phenomenon occurs in some specific areas compared with the intact slab; The type-Ⅱ stress intensity factor (SIF) is more sensitive to train speeds, while the support stiffness is highly correlated to all types of SIFs. This work may contribute to the scientific maintenance and health monitoring of high-speed railway tracks.

Physics-informed neural networks for approximating dynamic (hyperbolic) PDEs of second order in time: Error analysis and algorithms

We consider the approximation of a class of dynamic partial differential equations (PDEs) of second order in time by the physics-informed neural network (PINN) approach, and provide an error analysis of PINN for the wave equation, the nonlinear Klein-Gordon equation and the linear elastodynamic equation. Our analyses show that, with feed-forward neural networks having two hidden layers and the tanh activation function, the PINN approximation errors for the solution field, its time derivative and its gradient field can be effectively bounded by the training loss and the number of training data points (quadrature points). Our analyses further suggest new forms for the training loss function, which contain certain residuals that are crucial to the error estimate but would be absent from the canonical PINN loss formulation. Adopting these new forms for the loss function leads to a variant PINN algorithm. We present ample numerical experiments with the new PINN algorithm for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation, which show that the method can capture the solution well.

Dispersion analysis of SPH as a way to understand its order of approximation

Smoothed Particle Hydrodynamics (SPH) is a numerical method to solve dynamical partial differential equations (PDE). The basis of the method is a ¡¡kernel-based¿¿ way to compute the spatial derivatives of a function whose values are given in moving irregularly located nodes (Lagrangian particles). Accuracy of the SPH is determined by independent parameters — the shape of the kernel, the kernel size h, the distance between the particles Δx. Constructing high-order SPH-schemes for different types of PDE is a state-of-the-art problem of computational mathematics.For the classical SPH-approximation of one-dimensional hyperbolic equations (isothermal gas dynamics) we found that the order of approximation of smooth solution correlates to the dispersion properties of the method. To this end we analyzed the dispersion relation for the approximation and found analytical representation of the numerical wave phase velocity. Moreover, for the first time, the order of approximation with respect to Δx/h was confirmed in computational experiments on a dynamic problem of sound wave propagation. For two kernels with 2 and 4 continuum derivatives, the second and the fourth order of approximation, respectively, was found. This finding may be generalized as follows. The solution error in the one-dimensional case for a quasi-uniformly located particles has the form O((hλ)η+(Δxh)ξ), where ξ is a parameter determined by the shape of kernel (its smoothness, i.e. the number of continuum derivatives), η is a parameter that does not depend on the shape of kernel (for classical non-negative kernels η=2), λ is the wavelength.Our results indicates that to develop high-order SPH-schemes for hyperbolic equations besides improving the order of approximation with respect to h/λ one need to ensure the order of approximation with respect to Δx/h. To this end kernels of which smoothness is at least 4 are necessary.

Structural Vibration Suppression Using a Reduced-Order Extended State Observer-Based Nonsingular Terminal Sliding Mode Controller with an Inertial Actuator

In this paper, we mainly aimed to design a reduced-order extended state observer-based active vibration controller for a structural vibration control system with total disturbances, i.e., model uncertainties, higher harmonics, and external excitations. A reduced-order extended state observer (RESO)-based nonsingular terminal sliding mode vibration control (RESO–NTSMVC) method is proposed for the vibration suppression of an all-clamped plate structure with an inertial actuator. First, a second-order state space model of the thin plate, with an inertial actuator, was established by solving the dynamic partial differential equation and analyzing the physical model. Second, the total disturbances, i.e., model uncertainties, higher harmonics, and external excitations, were estimated and compensated for by using a RESO via a feedforward part. Third, a NTSMVC based on an estimated value was designed to obtain a fast-tracking rate and effective vibration suppression performance. In addition, the stability of the closed-loop system was proven by using a Lyapunov stability criterion. Finally, a semi-physical experimental instrument was built based on the MATLAB/Simulink real-time environment and the NI-PCIE6343 acquisition card to verify strong anti-disturbance performance and effective vibration control performance of the designed method. The experimental comparison results showed that the vibration amplitudes of the proposed method could be reduced by 11.7 dB, when the traditional extended state observer-based nonsingular terminal sliding mode vibration control (ESO–NTSMVC) method achieved a control effect of only 6.5 dB. The comparative experimental results showed that the proposed method possessed better vibration suppression performance and anti-disturbance performance.

Bayesian modeling of pattern formation from one snapshot of pattern.

Partial differential equations(PDEs) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a method, referred to as Bayesian modeling of PDEs (BM-PDEs), to estimate the best dynamical PDE for one snapshot of a objective pattern under the stationary state without ground truth. We apply BM-PDEs to nontrivial patterns, such as quasicrystals (QCs), a double gyroid, and Frank-Kasper structures. We also generate three-dimensional dodecagonal QCs from a PDE model. This is done by using the estimated parameters for the Frank-Kasper A15 structure, which closely approximates the local structures of QCs. Our method works for noisy patterns and the pattern synthesized without the ground-truth parameters, which are required for the application toward experimental data.

Nonlinear dynamic analysis of thermally deformed beams subjected to uniform loading resting on nonlinear viscoelastic foundation

The purpose of this article is to examine the dynamic behavior of shear deformation beams subjected to high-speed thermal and mechanical loadings. The structure's theoretical equations are derived by employing the first-order shear deformation theory and nonlinear strain-displacement relationships. Loading is accomplished in two different time stages. Initially, the beam is subjected to stress and strain due to fast surface heating. We explore the circumstances that result in dynamic bending or buckling. After that, the bent beam is abruptly exposed to consistent pressure over time. For this reason, two-parameter mechanical loading is proposed. The first parameter indicates the final loading amount, while the second specifies the loading rate. As a consequence of this load, heat-induced displacements are decreased. Additionally, it is investigated whether mechanical loading results in snap-through in the structure based on the Budiansky criterion. The transient heat conduction equation is solved analytically, and the temperature profile is obtained over time. The nonlinear dynamic partial differential equations are then solved by the Chebyshev collocation, Newmark, and Newton-Raphson numerical methods. Moreover, the effect of a nonlinear viscoelastic foundation on the beam response is examined in this research. After demonstrating validation, some novel outcomes are provided to characterize the system's response in various situations.

Reconstructing Dynamic 3D Models with Small Data by Integrating Position-Based Dynamics and PDE-Based Modelling

Simulation with position-based dynamics is very popular due to its high efficiency. However, it has the weaknesses of lacking details when too few vertices are involved in simulation and inefficiency when too many vertices are used for simulation. To tackle this problem, in this paper, we propose a new method of reconstructing dynamic 3D models with small data. The core elements of the proposed approach are a curve-represented geometric model and a physics-based mathematical model defined by dynamic partial differential equations. We first use the simulation method of position-based dynamics to generate a group of keyframe poses, which are used to create the deformation animation of a 3D model. Then, wireframe curves are extracted from skin deformation shapes of the 3D model at different keyframe poses. A physics-based mathematical model defined by dynamic partial differential equations is proposed. Its closed-form solution is obtained to represent the extracted curves, which are used to reconstruct the deformation models at different keyframe poses. Experimental examples and comparisons made in this paper indicate that the proposed method of reconstructing dynamic 3D models can greatly reduce data size while keeping good details.

Study on seismic response characteristics of the interlayer isolation structure

Based on Timoshenko beam theory and considering the layer bending deformation and shear deformation of the structure, a semi analytical solution method (SAM) is proposed for the interlayer isolation structure system. The equivalent mechanical model of cantilever beam is established by using the distributed parameter system, on which the dynamic partial differential equation is derived; the boundary condition and continuity condition are used to solve the equation to obtain the natural vibration characteristics; the orthogonal conditions are deduced by Betti’s law and the seismic response of the structure can be solved by the mode superposition method. A 10 story isolated structure is designed as well as analysed by the semi analytical method (SAM) and the ETABS finite element simulation (FESE) separately. The results show that: the existence of the isolation layer may reduce the response of the superstructure and amplify the response of the substructure, and the amplification can be solved by increasing the mass, bending stiffness of the superstructure and shear stiffness of the isolation layer. In regular interlayer isolation structure, the optimal frequency ratio of the superstructure and substructure increases with the rising of isolation layer position. The results also prove the correctness and accuracy of the SAM, which enriches the analysis theory of interlayer isolation structure and provides reference for the accurate analysis of structural response in engineering.

The generalized (2 + 1) and (3+1)-dimensional with advanced analytical wave solutions via computational applications

The analytical solutions for an important generalized Nonlinear evolution equations NLEEs dynamical partial differential equations (DPDEs) that involve independent variables represented by the (2 + 1)-dimensional breaking soliton equation, the (2 + 1)-dimensional Calogero--Bogoyavlenskii--Schiff  (CBS) equation, and the (2 +1)-dimensional Bogoyavlenskii's breaking soliton equation (BE), and some new exact propagating solutions to a generalized (3+1)-dimensional KP equation with variable coefficients are constructed by using a new algorithm of the first integral method (NAFIM) and determined some analytical solutions by appointing special values of the parameters. In addition to that, we showed a new variety and unique travelling wave solutions by graphical illustration with symbolic computations.

Deep learning of dynamic subsurface flow via theory-guided generative adversarial network

Generative adversarial network (GAN) has been shown to be useful in various applications, such as image recognition, text processing and scientific computing, due its strong ability to learn complex data distributions. However, the ability of standard GAN to process dynamic data is limited. In this study, a theory-guided generative adversarial network (TgGAN) is proposed to solve dynamic partial differential equations (PDEs). Different from standard GANs, the training term is no longer the true data and the generated data, but rather their residuals. In addition, such theories as governing equations and other physical constraints are encoded into the loss function of the generator to ensure that the prediction does not only honor the training data, but also obey these theories. TgGAN is proposed for dynamic subsurface flow with heterogeneous model parameters, and the data at each time step are treated as a two-dimensional image. In this study, several numerical cases are introduced to test the performance of the TgGAN. Predicting the future response, label-free learning and learning from noisy data can be realized easily by the TgGAN model, and the effects of the number of training data and the collocation points are also discussed. In order to improve the efficiency of TgGAN, the transfer learning algorithm is also employed. Moreover, the sensitivity of TgGAN to the hydraulic conductivity field is studied. Numerical results demonstrate that the TgGAN model is both robust and reliable for deep learning of dynamic PDEs.

Singularities of Solutions of Hamilton\u2013Jacobi Equations

This is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton–Jacobi equations in the past decades. We also review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations.

Continuum Modeling and Vibration Analysis of Cable-Harnessed Plate Structures of Periodic Patterns

Abstract Over the past decade, modeling of cable-harnessed space structures has received special attention due to the need for better accuracies than the existing models. As these structures become more lightweight upon the advancements in the materials science, it is imperative to further consider accurate models in which the dynamic effects of the added cables are better accounted for. Researchers have heavily focused on creating models for cable-harnessed beam-like structures, while very few studies have considered plate-like structures. The proposed research aims at the development of an analytical model for cable-harnessed plate-like structures. Cables are assumed to be periodic in geometry to allow for the application of an energy-equivalent homogenization technique. To begin with, a linear displacement field and a second-order Green-Lagrange strain tensor for strain–displacement relationships are considered. The strain and kinetic energies of the fundamental element are found using these relations. The repeated pattern of the fundamental element over the area of the plate structure allows for the employment of the homogenization approach in which the kinetic and strain energies per area of the fundamental element are found and assumed to remain the same as an equivalent homogenized solid plate-like element. The governing dynamic partial differential equations (PDEs) are found using the Hamilton's principle. The results are validated using the finite element analysis. A detailed parametric analysis is also performed to investigate the effects of various cable parameters and wrapping patterns on the dynamics of the host structure.

Certain class of non-Bazilević functions associated with exponential function

Its well-known that a Hankel matrix is one whose entries of the reverse diagonals are constant, i.e. Mathematician, physicists and engineers are attracted to this matrix because of their appearances and computational properties in different areas: dynamical systems, quantum mechanics, and partial differential equations. The main object in this paper is give an upper bound for the determinant of the third Hankel matrix for which the entries are belong to a new certain class of non-Bazilević functions in the open disk associated with exponential function.

Modal-Based Nonlinear Model Predictive Control for 3-D Very Flexible Structures

In this article a novel nonlinear model predictive control (NMPC) scheme is derived, which is tailored to the underlying structure of the intrinsic description of geometrically exact nonlinear beams (in which velocities and strains are primary variables). This is an important class of partial differential equation (PDE) models whose behavior is fundamental to the performance of flexible structural systems (e.g., wind turbines, high-altitude long-endurance aircraft). Furthermore, this class contains the much-studied Euler–Bernoulli and Timoshenko beam models, but has significant additional complexity (to capture 3-D effects and arbitrarily large displacements) and requires explicit computation of rotations in the PDE dynamics to account for orientation-dependent forces. A challenge presented by this formulation is that uncontrollable modes necessarily appear in any finite dimensional approximation to the PDE dynamics. We show, however, that an NMPC scheme can be constructed in which the error introduced by the uncontrollable modes can be explicitly controlled. It is demonstrated that the asymptotic error can be made insignificant (from a practical perspective) using our NMPC scheme and excellent performance is obtained even when applied to a highly resolved numerical simulation of the PDEs. We also present a generalization of Kelvin–Voigt damping to the intrinsic description of geometrically exact beams. Finally, special emphasis is placed on constructing a framework suitable for real-time NMPC control, where the particular structure of the underlying PDEs is exploited to obtain both efficient finite-dimensional models and numerical schemes.

Boundary Control of Nonlinear ODE/Wave PDE Systems With a Spatially Varying Propagation Speed

We consider the boundary control of a nonlinear ordinary differential equation (ODE) actuated through a wave equation whose propagation speed is spatially varying. The ODE state is driven by the uncontrolled boundary of the wave equation. We design a nonlinear backstepping compensator to enable global asymptotic stability of the closed-loop system. We deduce the controller design and the stability proof by introducing a two-step backstepping transformation. The first transformation recasts the original system into a coupled 2×2 first-order hyperbolic system with spatially varying coefficients cascading into a nonlinear ODE. The second transformation is used in the design of a compensator for the resulting cascaded system. Our design offers a global stability result that is guaranteed assuming that the spatially varying propagation speed is continuously differentiable and positive. Moreover, for nonlinear systems, our result is the first contribution enabling actual compensation of actuator delays governed by a coupled first-order hyperbolic partial differential equation (PDE) induced by a wave PDE dynamics with a spatially varying propagation speed. The validity of the proposed controller is illustrated by the benchmark system controlled via a cable.

Laser Sintering Control for Metal Additive Manufacturing by PDE Backstepping

Metal additive manufacturing (AM) has been intensively advanced due to numerous industrial applications, such as automobiles, aerospace, consumer electronics, and medical devices. The dynamics of the melt pool via laser sintering for metal AM has been studied by means of the thermodynamic phase change model known as the “Stefan problem”. In this article, we develop a control design for the laser power to drive the depth of the melt pool to the desired set point. The governing equation is described by a partial differential equation (PDE) defined on a time-varying spatial domain, which is dependent on the PDE state, and the optical penetration of the laser energy affects the PDE dynamics in the domain as well as at the surface boundary. First, we design the full-state feedback control law utilizing the entire spatial profile of the temperature in the melt pool and the moving interface position. The closed-loop system is proven to satisfy some conditions to validate the physical model, and its origin is shown to be exponentially stable. Next, we propose an observer-based output feedback control law by reconstructing the temperature profile with the availability of only the measured interface position and prove the analogous properties of the closed-loop system. Numerical simulation for a controller designed on a single-phase Stefan model is conducted on a more complex and realistic two-phase Stefan model, which incorporates the cooling effect from the solid phase. In addition, a bias in the interface location measurement is considered. The numerical results illustrate the robustness of the proposed feedback. By lowering the initial temperature in the solid and by increasing the interface sensor bias to more extreme levels, which leads to the controller’s failure (where the failure is exhibited through the entire metal freezing and the melt pool disappearing), we explore the limits of how much uncertainty our control law can handle.

Dynamic response of a three-beam system with intermediate elastic connections under a moving load/mass-spring

The objective of this research is to study the dynamic response characteristics of a three-beam system with intermediate elastic connections under a moving load/mass-spring. In this study, the finite Sine-Fourier transform was performed for the dynamic partial differential equations of a simply supported three-beam system (SSTBS) under a moving load and a moving mass-spring, respectively. The dynamic partial differential equations were transformed into dynamic ordinary differential equations relative to the time coordinates, and the equations were solved and the displacement Fourier amplitude spectral expressions were obtained. Finally, based on finite Sine-Fourier inverse transform, the expressions for dynamic response of SSTBS under the moving load and moving mass-spring were obtained. The proposed method, along with ANSYS, was used to calculate the dynamic response of the SSTBS under a moving load/mass-spring at different speeds. The results obtained herein were consistent with the ANSYS numerical calculation results, verifying the accuracy of the proposed method. The influence of the load/mass-spring’s moving speed on the dynamic deflections of SSTBS were analyzed. SSTBS has several critical speeds under a moving load/mass-spring. The vertical acceleration incurred by a change in the vertical speed of SSTBS due to the movement of mass-spring and the centrifugal acceleration produced by the movement of massspring on the vertical curve generated by SSTBS vibration could not be neglected.