Galerkin Projection Method Research Articles

Feature-based manifold modeling for the quasiperiodic wake dynamics of a pair of side-by-side cylinders

We propose a feature-based manifold modeling (FeMM) framework for the quasiperiodic wake dynamics of a pair of side-by-side cylinders. The key enabler is to embed the most parsimonious mean-field manifold based on the extracted features, such as force coefficients and probing data from experiments and numerical simulations. The manifold model is then identified under the mean-field constraints of the model structure, ensuring human-interpretability. The FeMM method is demonstrated with a two-dimensional incompressible flow crossing a pair of side-by-side cylinders, exhibiting a flip-flopping wake in quasiperiodic behavior. The transient and post-transient dynamics are characterized by two coupled oscillators associated with vortex shedding and gap flow oscillations. Dynamic mode decomposition analysis reveals significant modal interactions between these two flow mechanisms, posing a serious challenge to projection-based modeling approaches, such as the Galerkin projection method. Nevertheless, the FeMM approach, based on force measurements, yields an interpretable model that accounts for the mechanisms underlying the quasiperiodic dynamics, demonstrating its applicability to higher-order dynamics with multiple scales and invariant sets. This approach is expected to have broad applicability in dynamic modeling and state estimation in various real-world scenarios.

Sampling free iterative PCE filter for state and parameter estimation of nonlinear dynamical systems

We present a novel filter for state and parameter estimation in non-linear dynamical systems, based on a generalised Kalman filter formulation. To achieve a sampling-free implementation, polynomial chaos expansion (PCE) and a Galerkin projection method are utilized for the propagation of uncertainties through the system dynamics. The non-linear dynamics of the system are then linearised by a sequence of Gauss-Newton iterations in combination with linear Kalman updates. Additionally, we introduce a new square root implementation of the PCE-based filter. The proposed filter is evaluated on the Lorenz-63 and Lorenz-84 models for the task of simultaneous state and parameter estimation and is compared with two related approaches. Finally, the computational complexity of our square-root implementation is compared against two existing square root approaches.

Stochastic simulation of magnetically controlled convection in porous media with random porosity via Karhunen-Loève expansion and intrusive polynomial chaos expansion

Understanding and quantifying uncertainty factors are crucial for accurately predicting thermomagnetic convection phenomena. This study presented a mathematical model and algorithm framework for uncertainty analysis of thermomagnetic convection in porous media with random porosity. The proposed method combined Karhunen-Loève expansion and intrusive polynomial chaos expansion to represent the input random parameters and output response, respectively. The Galerkin projection method was employed to decouple the stochastic governing equations of thermomagnetic convection into deterministic governing equations that can be efficiently solved using the finite volume method. By solving the polynomial chaos expansion of the output response and employing the stochastic projection method to solve the associated decoupled governing equations, the temporal evolution and statistical characteristics of the output response were obtained. The study revealed that porosity uncertainty in the porous medium affects the thermomagnetic convection of paramagnetic fluid, exhibiting significant chaos effects. Monte Carlo simulations were performed to validate the accuracy of the proposed approach and demonstrate its computational efficiency compared to traditional Monte Carlo methods. This research provides valuable insights into the uncertainty analysis of thermomagnetic convection and offers a promising methodology for analyzing heat transfer in various porous media applications.

Reduced-Order Model Approaches for Predicting Airfoil Performance

This study delves into the construction of reduced-order models (ROMs) of a flow field over a NACA 0012 airfoil at a moderate Reynolds number and an angle of attack of 8∘. Numerical simulations were computed through the finite-volume solver OpenFOAM. The analysis considers two different reduction techniques: the standard Galerkin projection method, which involves projecting the governing equations onto proper orthogonal decomposition modes (POD−ROMs), and the cluster-based network model (CNM), a fully data-driven nonlinear approach. An analysis of the topology of the dominant POD modes was conducted, uncovering a traveling wave pattern in the wake dynamics. We compared the performances of both ROM techniques regarding their prediction of flow field behavior and integral quantities. The ROM framework facilitates the practical actuation of control strategies with significantly reduced computational demands compared to the full-order approach.

Gyroscopically coupled in-plane and transverse vibrations of an annular free-clamped microplate

In this work, we construct and study a model of coupled plane-transverse oscillations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertia forces caused by the rotation of the system along an axis located in the plane of the plate. Equations of vibrations in partial derivatives are obtained using the variational principle of Hamilton - Ostrogradsky. Assuming the smallness of the angular velocity of rotation with respect to the frequency of the working skew-symmetric flexural form of the plate oscillations, an approximate analytical solution is found for both the radial and circumferential, and transverse components of the displacement field in the free oscillation mode. Using the Galerkin projection method, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric modes of the plate vibrations. It is found that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into the regime of amplitude-modulated beats. Analytical expressions are found both for the frequency of the slow beat envelope and for the relative depth of their amplitude modulation. The fundamental possibility of determining the modulus of the projection of the angular velocity vector onto the plane of the plate from the measured value of the envelope frequency is shown. The problem of choosing the optimal geometric shape of the resonator from the point of view of maximizing the sensitivity of the system to changes in the value of the angular velocity of rotation is studied. The question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of the amplitude modulation of the beat regime is considered.

A splitting discontinuous Galerkin projection method for the magneto-hydrodynamic equations

This paper is focused on the numerical approximations of the incompressible magneto-hydrodynamic equations. Our interest is to develop a novel decoupled, linear and unconditional energy stable fully-discrete scheme, which is achieved by the interior penalty discontinuous Galerkin (DG) method for spatial discretization, the stabilizing strategy and implicit-explicit (IMEX) scheme used to handle the nonlinear coupling terms, and a rotational pressure-correction method for the Navier-Stokes equations. We prove the unique solvability, unconditional energy stability and optimal error estimates of the proposed scheme rigorously. We further present several numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme numerically.

Efficient interior penalty discontinuous Galerkin projection method with unconditional energy stability and second-order temporal accuracy for the incompressible magneto-hydrodynamic system

In this paper, we propose a novel interior penalty discontinuous Galerkin projection method for the incompressible magneto-hydrodynamic equations. The scheme is employed by an implicit-explicit treatment of the nonlinear coupling terms and a second-order rotational pressure-correction scheme for dealing with the Navier-Stokes equations. One noteworthy aspect of this scheme is the introduction of an additional stabilization term to Maxwell's equations, which allows for the explicit treatment of the coupled nonlinear terms and the decoupling of computations for the magnetic and velocity fields, ultimately leading to the achievement of desired linearity, full decoupling, second-order accuracy in time, and unconditional energy stability. The obtained scheme is easy to implement because one only needs to solve a few decoupled linear equations at each time step. We rigorously prove the unique solvability and unconditional energy stability of the developed scheme and present a series of numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme.

Nonunit quaternion parametrization of a Petrov–Galerkin Cosserat rod finite element

AbstractThe application of the Petrov–Galerkin projection method in Cosserat rod finite element formulations offers significant advantages in simplifying the expressions within the discrete virtual work functionals. Moreover, it enables a straight‐forward and systematic exchange of the ansatz functions, specifically for centerline positions and cross‐section orientations. In this concise communication, we present a total Lagrangian finite element formulation for Cosserat rods that attempts to minimize the number of required theoretical concepts. The chosen discretization preserves objectivity and allows for large displacements/rotations and for large strains. The orientation parametrization with nonunit quaternions results in a singularity‐free formulation.

Nonlinear reduced-order modeling for three-dimensional turbulent flow by large-scale machine learning

A large-scale machine learning-based nonlinear reduced-order modeling method was developed for a three-dimensional turbulent flow field (Re=1000) using a neural-network with unsupervised learning. First, a mode decomposition method was applied to three-dimensional flow field data using a convolutional-autoencoder-like neural network. Then, a reduced-order model (ROM) was constructed using long short-term memory neural networks (LSTMs). Consequently, it was demonstrated that the time evolution of the turbulent three-dimensional flow field can be simulated at a significantly lower cost (approximately three orders of magnitude) without a major loss in accuracy. However, neural-network-based mode decomposition for the three-dimensional flow field requires a huge computational cost in terms of calculation and memory usage. Therefore, a distributed machine-learning method was implemented using a hybrid parallelism scheme tailored to the network structure. Thus, it was possible to decompose 1.7 million cells of the three-dimensional flow field data into 64 modes and reproduce those with sufficient accuracy. In this study, a uniform flow around a circular cylinder model was used as a test case. To validate the method, the reduction performance of the proposed mode decomposition method was compared with the proper orthogonal decomposition (POD) method. Furthermore, the target flow field was reproduced using ROM, and the reconstruction accuracy was evaluated in terms of various criteria compared with the accuracy based on POD in conjunction with Galerkin projection method.

Propagation algorithm for hybrid uncertainty parameters based on polynomial chaos expansion

AbstractThis article presents an uncertainty analysis method for systems with hybrid stochastic and fuzzy uncertainty parameters based on polynomial chaos expansion (PCE). Parameters in the system are described by probability boxes, interval numbers, and fuzzy sets, respectively, based on the differences in their limited stochastic knowledge. First, this method transforms the uncertain parameters into standard normal distribution and interval variables through equal probability transformation or ‐cut operations. Second, the Legendre and Hermite polynomials are used as the PCE model's primary functions, and the expansion coefficients are calculated by the Galerkin projection method based on sparse grid numerical integration. Then, the system response bounds under the pre‐defined confidence level can be obtained using a genetic algorithm to solve the optimization problem constructed based on PCE models. Finally, the feasibility and effectiveness of the method are illustrated by taking the tank bi‐directional stabilized system and the double‐pendulum‐controlled system as examples. The numerical results show that the system response bounds obtained by the PCE model optimization algorithm are consistent with the Monte Carlo simulation. Still, the computational efficiency is much higher. The proposed method effectively combines fuzzy sets and probability boxes and dramatically simplifies the analysis process of uncertain systems. The method exhibits fine precision even in high‐dimensional uncertainty analysis problems.

A family of total Lagrangian Petrov–Galerkin Cosserat rod finite element formulations

SummaryThe standard in rod finite element formulations is the Bubnov–Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross‐section orientations. Using a Petrov–Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross‐section orientations. We discuss three objective interpolation strategies and give an in‐depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.

A total Lagrangian, objective and intrinsically locking‐free Petrov–Galerkin SE(3) Cosserat rod finite element formulation

AbstractBased on more than three decades of rod finite element theory, this publication combines the successful contributions found in the literature and eradicates the arising drawbacks like loss of objectivity, locking, path‐dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jelenić in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the ‐structure of the Cosserat rod kinematics. Applying a Petrov–Galerkin projection method, we propose a rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time‐derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parameterization in terms of a minimal number of nodal unknowns are demonstrated by representative numerical examples in both statics and dynamics.

BUQEYE guide to projection-based emulators in nuclear physics

The BUQEYE collaboration (Bayesian Uncertainty Quantification: Errors in Your effective field theory) presents a pedagogical introduction to projection-based, reduced-order emulators for applications in low-energy nuclear physics. The term emulator refers here to a fast surrogate model capable of reliably approximating high-fidelity models. As the general tools employed by these emulators are not yet well-known in the nuclear physics community, we discuss variational and Galerkin projection methods, emphasize the benefits of offline-online decompositions, and explore how these concepts lead to emulators for bound and scattering systems that enable fast and accurate calculations using many different model parameter sets. We also point to future extensions and applications of these emulators for nuclear physics, guided by the mature field of model (order) reduction. All examples discussed here and more are available as interactive, open-source Python code so that practitioners can readily adapt projection-based emulators for their own work.

Обобщение метода Петрова–Галеркина для решения системы интегральных уравнений Фредгольма второго рода

The numerical solution problem of a second kind Fredholm m integral equations linear system is considered. The Petrov–Galerkin projection method generalization for solving this problem is proposed the first time. n is a coordinate functions number of two linearly independent systems. It can be equal, more or less then the m – the integral equations number. This algorithm advantage is that it is not sensitive to the parameters λ smallness in the integral equations system. The algorithm requires the correct choice of two linearly independent coordinate functions systems and their number. The solution algorithm is reduced to a matrix solution for an antidiagonal problem. Two examples are solved for the antidiagonal problem, and numerical solutions of the problem coincide with the exact solutions. Two theorems are proved for sufficient conditions for the proposed numerical algorithms correctness in two cases. In the first case, an antidiagonal problem is considered with two Fredholm equations of the second kind. The second theorem considers well-posedness conditions for a general diagonal problem. Undoubtedly, the proposed algorithm will be useful in mechanics and computational mathematics problems.

One Line of Development of the Galerkin Projection Method in Problems of Stationary Solid Mechanics (Review)

One Line of Development of the Galerkin Projection Method in Problems of Stationary Solid Mechanics (Review)

Preconditioned least‐squares Petrov–Galerkin reduced order models

AbstractIn this article, we introduce a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least‐squares Petrov–Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual‐based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi‐physics problems). Importantly, we demonstrate that an “ideal preconditioned” LSPG ROM (a ROM in which the preconditioner is the inverse of the Jacobian of its corresponding full order model) emulates projection of the full order model solution increment onto the reduced basis. This quantity defines a lower bound on the error of a ROM solution for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse—as is common in designing preconditioners for solving linear systems—it is possible to obtain a ROM whose error approaches this lower bound. The proposed approach is evaluated on several mechanical and thermo‐mechanical problems implemented within the Albany HPC code and run in the predictive regime, with prediction across material parameter space. We demonstrate numerically that the introduction of simple Jacobi, Gauss‐Seidel, and ILU preconditioners into the proper orthogonal decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time (thereby improving efficiency). Moreover, our numerical results reveal that the introduction of preconditioning can deliver a robust and accurate solution for test cases in which the unpreconditioned LSPG method fails to converge.

A Galerkin/hyper-reduction technique to reduce steady-state elastohydrodynamic line contact problems

A novel model order reduction technique for the elastohydrodynamic problem consisting of the Reynolds equation and the elasticity equation is proposed that drastically reduces the dimensionality of the problem and the computing time. The method is developed for stationary Newtonian isothermal line contacts. It employs the Galerkin projection method for the structural part combined with a hyper-reduction technique for the Reynolds equation. The model order reduction strategy consists of two phases. A first offline phase where snapshots of the full order model solution are computed using a selective refinement technique; and an online phase where the precomputed bases are used to compute the reduced order model. The method leads to a large reduction in the dimensionality of the full order model which results in a considerable reduction of the computing time, in a o(103) speedup factor, with an excellent correlation to the results obtained with the full order model.

Bifurcation analysis of vortex-induced vibration of low-dimensional models of marine risers

A low-dimensional model of a top-tensioned riser under excitations from vortices and time-varying tension is proposed, where the van der Pol wake oscillator is used to simulate the loading caused by the vortex shedding. The governing partial differential equations describing the fluid–structure interactions are formulated and multi-mode approximations are obtained using the Galerkin projection method. The one mode approximation is applied in this study and two different resonances are investigated by employing the method of multiple scales. They are the 1:1 internal resonance between the structure and wake oscillator (also known as ‘lock-in’ phenomenon) and the combined 1:1 internal and 1:2 parametric resonances. Bifurcations under the varying nondimensional shedding frequency for different mass–damping parameters are investigated and the results of multiple-scale analysis are compared with direct numerical simulations. Analytical responses are calculated using the continuation method and their stability is determined by examining the eigenvalues of the corresponding characteristic equations. Effects of the system parameters including the amplitude of the tension variation, vortex shedding frequency and mass–damping parameter on the system bifurcations have been investigated. The analytical approach has allowed to probe bifurcations occurring in the system and to identify stable and unstable responses. It is shown that the combined resonances can induce large-amplitude vibration of the structure. Counter-intuitively, the amplitude of such responses increases rapidly as the amplitude of the tension variation grows. Comparisons between the analytical and numerical results confirm that the span of the system vibration can be accurately predicted analytically with respect to the obtained response amplitudes of responses. The proposed multi-mode approximation and presented findings of this study can be used to enhance design process of top tension risers.

Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise

In this paper, we investigate the numerical approximation of stochastic convection–reaction–diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately 1 for trace class noise and for space‐time white noise. These convergence orders are obtained under less regularity assumptions on the nonlinear drift function than those used in the literature for stochastic reaction–diffusion equations. Numerical experiments to illustrate our theoretical results are provided.

Asymmetrical Barrier Function-Based Adaptive Vibration Control for Nonlinear Flexible Cantilever Beam With Obstacle Restriction

In this paper, a novel adaptive asymmetrical barrier function-based vibration control law is proposed for the nonlinear flexible cantilever beam system with obstacle restriction, model uncertainties, and distributed disturbances. Firstly, by employing the Hamilton’s principle and the Galerkin projection method, the dynamic of the nonlinear flexible cantilever beam with the piezoelectric actuator is constructed in partial differential equations and simplified to nonlinear ordinary differential equations for the control law design. Then, by introducing the fast nonsingular terminal sliding mode surface, a novel asymmetrical barrier function based sliding mode control law is proposed, in which by means of a novel asymmetrical barrier Lyapunov function is used to guarantee the finite time stability with the obstacle restriction. Further, to deal with the model uncertainties, an adaptive updating law is incorporated with the fast nonsingular terminal sliding mode control law based on asymmetrical barrier function, stability proof shows that the proposed control law can ensure the distributed disturbance rejection and the model uncertainties compensation simultaneously. Finally, the effectiveness of the proposed control laws is demonstrated by the numerical simulations.