Galerkin Model Research Articles

Windowed space–time least-squares Petrov–Galerkin model order reduction for nonlinear dynamical systems

Projection-based reduced-order models (PROMs) restrict the full-order model (FOM) to a low-dimensional subspace. Space–time PROMs in particular restrict the FOM to a temporally space–time trial subspace, and compute approximate solutions through a residual orthogonalization or minimization process. One such technique of interest is the space–time least-squares Petrov–Galerkin method (ST-LSPG), which reduces both the spatial and temporal dimensions. However, ST-LSPG is computationally expensive, because it requires solving a dense space–time system with a space–time basis that is calculated over the entire global time domain, which can be unfeasible for large-scale applications. To address these challenges, this paper presents the windowed space–time least-squares Petrov–Galerkin method (WST-LSPG) for model reduction of nonlinear parameterized dynamical systems. The proposed WST-LSPG approach addresses the aforementioned challenges by (1) dividing the time simulation into windows, (2) devising a unique low-dimensional high-fidelity space–time trial subspace for each window, and (3) minimizing the discrete-in-time space–time residual of the dynamical system over each window. In this formulation, the problem confines coupling within each window, but solves space–time residual minimization problems sequentially across the windows. WST-LSPG is equipped with hyper-reduction techniques to further reduce the computational cost. Numerical experiments for the one-dimensional Burgers’ equation and the two-dimensional compressible Navier–Stokes equations for flow over a NACA 0012 airfoil demonstrate that WST-LSPG is superior to ST-LSPG in terms of accuracy and computational gain by as much as 99%.

Physics-constrained, low-dimensional models for magnetohydrodynamics: First-principles and data-driven approaches.

Plasmas are highly nonlinear and multiscale, motivating a hierarchy of models to understand and describe their behavior. However, there is a scarcity of plasma models of lower fidelity than magnetohydrodynamics (MHD), although these reduced models hold promise for understanding key physical mechanisms, efficient computation, and real-time optimization and control. Galerkin models, obtained by projection of the MHD equations onto a truncated modal basis, and data-driven models, obtained by modern machine learning and system identification, can furnish this gap in the lower levels of the model hierarchy. This work develops a reduced-order modeling framework for compressible plasmas, leveraging decades of progress in projection-based and data-driven modeling of fluids. We begin by formalizing projection-based model reduction for nonlinear MHD systems. To avoid separate modal decompositions for the magnetic, velocity, and pressure fields, we introduce an energy inner product to synthesize all of the fields into a dimensionally consistent, reduced-order basis. Next, we obtain an analytic model by Galerkin projection of the Hall-MHD equations onto these modes. We illustrate how global conservation laws constrain the model parameters, revealing symmetries that can be enforced in data-driven models, directly connecting these models to the underlying physics. We demonstrate the effectiveness of this approach on data from high-fidelity numerical simulations of a three-dimensional spheromak experiment. This manuscript builds a bridge to the extensive Galerkin literature in fluid mechanics and facilitates future principled development of projection-based and data-driven models for plasmas.

Machine learning-based model predictive control of diffusion-reaction processes

In this work, we develop a machine-learning-based predictive control design for nonlinear parabolic partial differential equation (PDE) systems using process state measurement time-series data. First, the Karhunen-Loève expansion is used to derive dominant spatial empirical eigenfunctions of the nonlinear parabolic PDE system from the data. Then, these empirical eigenfunctions are used as basis functions within a Galerkin's model reduction framework to derive the temporal evolution of a small number of temporal modes capturing the dominant dynamics of the PDE system. Subsequently, feedforward neural networks (FNN) are used to approximate the reduced-order dominant dynamics of the parabolic PDE system from the data within a desired operating region. Lyapunov-based model predictive control (MPC) scheme using FNN models is developed to stabilize the nonlinear parabolic PDE system. Finally, a diffusion-reaction process example is used to demonstrate the effectiveness of the proposed machine-learning-based predictive control method.

A compute-bound formulation of Galerkin model reduction for linear time-invariant dynamical systems

This work aims to advance computational methods for projection-based reduced-order models (ROMs) of linear time-invariant (LTI) dynamical systems. For such systems, current practice relies on ROM formulations expressing the state as a rank-1 tensor (i.e., a vector), leading to computational kernels that are memory bandwidth bound and, therefore, ill-suited for scalable performance on modern architectures. This weakness can be particularly limiting when tackling many-query studies, where one needs to run a large number of simulations. This work introduces a reformulation, called rank-2 Galerkin, of the Galerkin ROM for LTI dynamical systems which converts the nature of the ROM problem from memory bandwidth to compute bound. We present the details of the formulation and its implementation, and demonstrate its utility through numerical experiments using, as a test case, the simulation of elastic seismic shear waves in an axisymmetric domain. We quantify and analyze performance and scaling results for varying numbers of threads and problem sizes. Finally, we present an end-to-end demonstration of using the rank-2 Galerkin ROM for a Monte Carlo sampling study. We show that the rank-2 Galerkin ROM is one order of magnitude more efficient than the rank-1 Galerkin ROM (the current practice) and about 970 times more efficient than the full-order model, while maintaining accuracy in both the mean and statistics of the field.

Galerkin force model for transient and post-transient dynamics of the fluidic pinball

Abstract

Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup

This paper presents a new depth-integrated non-hydrostatic finite element model for simulating wave propagation, breaking and runup using a combination of discontinuous and continuous Galerkin methods. The formulation decomposes the depth-integrated non-hydrostatic equations into hydrostatic and non-hydrostatic parts. The hydrostatic part is solved with a discontinuous Galerkin finite element method to allow the simulation of discontinuous flows, wave breaking and runup. The non-hydrostatic part led to a Poisson type equation, where the non-hydrostatic pressure is solved using a continuous Galerkin method to allow the modeling of wave propagation and transformation. The model uses linear quadrilateral finite elements for horizontal velocities, water surface elevations and non-hydrostatic pressures approximations. A new slope limiter for quadrilateral elements is developed. The model is verified and validated by a series of analytical solutions and laboratory experiments.

Comparison of Taylor Galerkin and FTCS models for dam-break simulation

The aim of this paper is to compare the ability of two numerical models to simulate two-dimensional dam-break flows. Dam-break modelling is important in reducing the high-risk of dam failure by determining the water flow movement and its effect on buildings downstream. In this study, the Taylor Galerkin and Forward Time Central Space method is presented to simulate 2-D dam-break flow. The Hansen filter is applied to both methods to reduce numerical instability oscillations in the model. The filters were used for each time step at each computing point. The model is verified by comparing the result simulation with numerical results from previous studies. The result show that Taylor Galerkin numerical scheme with Hansen filter is successful to simulate the problem of 2-D dam-break. Also, the Taylor Galerkin model has better results than the FTCS model. Studies like this are needed for mitigation efforts in the event of dam failure.

Calibration of projection-based reduced-order models for unsteady compressible flows

An analysis of calibration for reduced-order models (ROMs) is presented in this work. The Galerkin and least-squares Petrov-Galerkin (LSPG) methods are tested on compressible flows involving a disparity of temporal scales. A novel calibration strategy is proposed for the LSPG method and two test cases are analyzed. The first consists of a subsonic airfoil flow where boundary layer instabilities are responsible for trailing-edge noise generation and the second comprises a supersonic airfoil flow with a transient period where a detached shock wave propagates upstream at the same time that shock-vortex interaction occurs at the trailing edge. Results show that calibration produces stable and long-time accurate Galerkin and LSPG ROMs for both cases investigated. The impact of hyper-reduction is tested on LSPG models via an accelerated greedy missing point estimation (MPE) algorithm. For the first case investigated, LSPG solutions obtained with hyper-reduction show good comparison with those obtained by the full order model. However, for the supersonic case the transient features of the flow need to be properly captured by the sampled points of the accelerated greedy MPE method. Otherwise, the dynamics of the moving shock wave are not fully recovered. The impact of different time-marching schemes is also assessed and, differently than reported in literature, Galerkin models are shown to be more accurate than those computed by LSPG when the non-conservative form of the Navier-Stokes equations is solved. For the supersonic case, the Galerkin and LSPG models (without hyper-reduction) capture the overall dynamics of the detached and oblique shock waves along the airfoil. However, when shock-vortex interaction occurs at the trailing-edge, the Galerkin ROM is able to capture the high-frequency fluctuations from vortex shedding while the LSPG presents a more dissipative solution, not being able to recover the flow dynamics.

A Reduced Order Controller for Output Tracking of a Kelvin–Voigt Beam

We study output tracking and disturbance rejection for an Euler-Bernoulli beam with Kelvin-Voigt damping. The system has distributed control and pointwise observation. As our main result we design a finite-dimensional low-order internal model based controller that is based on a spectral Galerkin method and model reduction by Balanced Truncation. The performance of the designed controller is demonstrated with numerical simulations and compared to the performance of a low-gain internal model based controller.

POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation

In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

Frequency-dependent transmission boundary condition in the acoustic time-domain nodal discontinuous Galerkin model

Accurate modeling of boundary conditions is of critical importance for acoustic simulations. Recently, the time-domain nodal discontinuous Galerkin (TD-DG) method has emerged as a potential wave-based method for acoustic modeling. Although the acoustic reflection behavior of various time-domain impedance boundaries has been studied extensively, the modeling of the sound transmission across a locally-reacting layer of impedance discontinuity is far less developed. This paper presents a formulation of broadband time-domain transmission boundary conditions for locally-reacting surfaces in the framework of the TD-DG method. The formulation simulates the acoustic wave behavior at each of the boundary nodes using the plane-wave theory. Through the multi-pole model representation of the transmission coefficient, various types of transmission layers can be simulated. One-dimensional numerical examples demonstrate the capability of the proposed formulation to accurately simulate the reflection and transmission characteristics of the limp wall and the porous layer, where quantitative error behavior against analytical results is presented. Furthermore, to demonstrate the applicability, two scenarios of two-dimensional acoustic environment are considered. One is the sound transmission between two rooms partitioned by a limp panel and the other is the sound propagation through a transmissive noise barrier. Comparison of the predicted results from the proposed method against the results from the frequency-domain finite element simulations further verifies the formulation.

Low-order model for successive bifurcations of the fluidic pinball

We propose the first least-order Galerkin model of an incompressible flow undergoing two successive supercritical bifurcations of Hopf and pitchfork type. A key enabler is a mean-field consideration exploiting the symmetry of the mean flow and the asymmetry of the fluctuation. These symmetries generalize mean-field theory, e.g. no assumption of slow growth-rate is needed. The resulting 5-dimensional Galerkin model successfully describes the phenomenogram of the fluidic pinball, a two-dimensional wake flow around a cluster of three equidistantly spaced cylinders. The corresponding transition scenario is shown to undergo two successive supercritical bifurcations, namely a Hopf and a pitchfork bifurcations on the way to chaos. The generalized mean-field Galerkin methodology may be employed to describe other transition scenarios.

Adaptive Discontinuous Galerkin Modeling of Intrinsic Attenuation Anisotropy for Fluid-Saturated Porous Media

An hp- and memory-adaptive discontinuous Galerkin time-domain algorithm is presented to efficiently model wave propagation in poroelastic media, with the incorporation of 3-D fully anisotropic intrinsic attenuation. From the perspective of physics, the attenuation from the triclinic rock frame, the loss in the pore fluid, and the friction for their interaction are all considered. From the perspective of implementation, a new frequency-domain constitutive equation is introduced, involving complex-valued poroelasticity matrix. A new $Q$ value is defined as the ratio between its real and imaginary parts for every entry. Then, a generalized Maxwell body is adopted to approximate this frequency-dependent $Q$ in the time domain. Mathematically speaking, the hyperbolicity is preserved for the new viscous poroelastic system. Our results corroborate that the intrinsic attenuation anisotropy makes tangible effects in fluid-saturated porous formations.

Depth-integrated wave–current models. Part 1. Two-dimensional formulation and applications

Depth-integrated mathematical models for simulating waves and currents from deep to shallow water are presented. These models are derived from Euler’s equations in the -coordinate system, mapping the total water depth in Cartesian coordinates onto a specified range of -coordinates. The horizontal velocity is approximated as a truncated infinite series of products of prescribed shape functions of and unknown functions of horizontal coordinates and time. Adopting the method of weighted residuals, the new models are obtained by minimizing the residuals of the horizontal momentum equations with either the Galerkin method or the subdomain method. These models’ linear and nonlinear water wave properties are investigated. The new models are implemented numerically. A hierarchy of numerical models with different degree of polynomial approximation is developed and checked against several benchmarked experiments and a new set of experiments of self-focusing wave groups. For both the Galerkin and subdomain models, excellent agreements are observed for both the free surface elevations and the velocity profiles. The new models are superior to the existing Boussinesq-type models for their applicability to a wide range of physical scenarios, including the interactions between a wave package of multiple frequency components and a linearly sheared current. The new Galerkin models have similar characteristics and accuracy as the Green–Naghdi models, but the new models are more efficient computationally. Finally, for the same degree of polynomial approximation the subdomain models perform better than the Galerkin models and require less computational time.

Tunneling of persistent currents in coupled ring-shaped Bose–Einstein condensates

Considerable progress in experimental studies of atomic gases in a toroidal geometry opens up novel prospects for the investigation of fundamental properties of superfluid states and creation of new configurations for atomtronic circuits. In particular, a setting with Bose–Einstein condensates loaded in a dual-ring trap suggests a possibility to consider the dynamics of tunneling between condensates with different angular momenta. Accordingly, we address the tunneling in a pair of coaxial three-dimensional (3D) ring-shaped condensates separated by a horizontal potential barrier. A truncated (finite-mode) Galerkin model and direct simulations of the underlying 3D Gross–Pitaevskii equation are used for the analysis of tunneling superflows driven by an initial imbalance in atomic populations of the rings. The superflows through the corresponding Bose–Josephson junction are strongly affected by persistent currents in the parallel rings. Josephson oscillations of the population imbalance and angular momenta in the rings are obtained for co-rotating states and non-rotating ones. On the other hand, the azimuthal structure of the tunneling flow demonstrates formation of Josephson vortices (fluxons) with zero net current through the junction for hybrid states, built of counter-rotating persistent currents in the coupled rings.

Meshless Method for Nonuniform Heat-Transfer/Solidification Behavior of Continuous Casting Round Billet

Numerical simulation is the primary approach to evaluate the complex solidification behavior for the continuous casting (CC) process, in which the methods based on the mesh and topology technique have been widely used to solve field variables. For the traditional techniques, such as the finite difference, finite element, and boundary element methods, it is arduous or even impossible to deal with complicated problems that involve multiphase coupling, interface tracking/reconstruction, and self-adaptation due to the inherent weaknesses in the grid structure and mesh dependence. Hence, the present work explores a meshless calculation method for the two-dimensional unsteady heat-transfer problem and proposes an element-free Galerkin (EFG) model for solving heat transfer inside the CC mold based on the moving least-squares approximation. The temperature functions are approximated and constructed by a linear basis and cubic spline weight function over a set of rectangular supporting domain; then, the discrete heat-transfer governing equation based on the EFG method is deduced. The heat flux measured in the real casting process is set as the boundary condition to calculate the nonuniform solidification of the round billet. The calculated results demonstrate that the shell thickness is consistent with that obtained by the square root law of solidification. In addition, the high heat flux region near the meniscus directly determines the growth characteristics of the initial billet shell, which ultimately results in the overall nonuniformity of the shell. The EFG method has the characteristics of fast convergence, high computational accuracy, and great discrete flexibility; it also provides a novel and effective approach for subsequent thermomechanical coupling and crack propagation analysis in the CC process.

Feasibility study of a resonant accelerometer with bistable electrostatically actuated cantilever as a sensing element

We report on a feasibility study of a resonant accelerometer incorporating fringing electrostatic fields actuated cantilever serving as a sensing element. Device’s dynamics are described using the reduced order (RO) Galerkin and numerical finite differences (FD) models, the finite elements (FE) analysis is used for the evaluation of the electrostatic forces. The architecture of the electrodes designed to be thicker than the cantilever allows tuning of the beam behavior in a wide range, starting from linear and up to bistable responses. By choosing an appropriate value of the actuating voltage close to the bistability threshold the cantilever can be positioned in the configuration where the frequency sensitivity of the device to the electrode’s deflection is enhanced while the frequency itself is higher than in the initial inactuated state. In accordance with the model results, by integrating the bistable cantilever with a proof mass, it is possible to design a highly sensitive resonant accelerometer with the state of the art performance.

Micro-beam resonator parametrically excited by electro-thermal Joule’s heating and its use as a flow sensor

We study parametric resonance (PR) of double-clamped micro-beams that are electro-thermally actuated by a time-dependent Joule’s heating and cooled by a steady air flow. The developed model demonstrates applicability of such device as a bifurcation-based flow velocity sensor. An AC electric current through the beam induces a time-harmonic compressive force that leads to parametric excitation of the structure. Convective cooling due to the air flow affects the location of the parametric transition curves on the driving voltage–frequency plane. The flow velocity can be obtained by measuring the frequency corresponding to the steep amplitude transition of the response. The device is modeled as an Euler–Bernoulli beam with an axial force parameterized by the electric current. The heat transfer problem is solved analytically; the heat flux due to the air flow is calculated using empirical correlations. The behavior of the beam is studied numerically, by means of finite differences, and analytically, using an approximate single degree of freedom Galerkin model, reduced to the Mathieu–Duffing equation. We show that while the PR always emerges at the driving voltage/current below the critical static buckling value, practical realization of the purely electro-thermal parametric excitation is challenging and is highly influenced by the device dimensions and quality factors. We evaluate the parameters required to assure the PR and demonstrate, using the model, feasibility of the suggested flow-sensing approach in the devices of realistic dimensions.

Size-Dependent elastic response in functionally graded microbeams considering generalized first strain gradient elasticity

Summary In this article, the size-dependent mechanical response of an isotropic functionally graded (FG) microbeam has been investigated. The size-effects over the elastic response have been modeled by the Mindlin–Toupin strain gradient theory, with the coefficients evaluated from the generalized first strain gradient theory of elasticity. In order to facilitate the derivation of the exact solutions to the governing differential equations of equilibrium, an exponentially graded FG beam is chosen. These exact solutions are derived for a simply supported beam subjected to a sinusoidally distributed mechanical load. Following this, an element-free Galerkin (EFG) model involving moving least squares interpolations across the domain is also developed here. The EFG model is validated with the exact solutions for the exponentially graded beam. Finally, the EFG model is extended to the more general case of a power law-graded beam. The mechanical responses for the power law-graded beams under various loading and boundary conditions are presented here. These results may serve as benchmark for further studies over size-effects in FG beams.

New finite volume multiscale finite element model for simultaneously solving groundwater flow and darcian velocity fields in porous media

This paper proposes a new finite volume multiscale finite element model (NFVMSFEM) for groundwater flow and velocity simulation in porous media. The main goal of this method is to simultaneously obtain the heads and continuous x,y direction Darcian velocities, thereby simplifying the simulation process and improving the efficiency. This goal is accomplished by introducing a novel velocity matrix in the control volume boundary flux estimation of the finite volume multiscale finite element method (FVMSFEM) framework. The velocity matrix allows for the NFVMSFEM to directly represent continuous fine-scale velocities by coarse-scale heads, thus merging the two kinds of unknowns into one. Through a modified Yeh’s Galerkin model, the velocity matrix is constructed in each local coarse element before the head computation; thus, the computational cost is very little. Furthermore, the NFVMSFEM inherits the advantageous features of the FVMSFEM, which ensures the local conservation of heads and further improves the efficiency. The results of the comparison between the NFVMSFEM and certain classical methods for head and velocity computation demonstrate its effectiveness and high efficiency.