Data assimilation performance can be significantly impacted by biased noise in observations, altering the signal magnitude and introducing fast oscillations or discontinuities when the system lacks smoothness. To mitigate these issues, this paper employ variational state estimation using the so-called parametrized-background data-weak method. This approach relies on a background manifold parametrized by a set of constraints, enabling the state estimation by solving a minimization problem on a reduced-order background model, subject to constraints imposed by the input measurements. The proposed formulation incorporates a novel bias correction mechanism and a manifold decomposition that handles rapid oscillations by treating them as slow-decaying modes based on a two-scale splitting of the classical reconstruction algorithm. The method is validated in different examples, including the assimilation of biased synthetic data, discontinuous signals, and Doppler ultrasound data obtained from experimental measurements.
In this paper, we develop data-driven closure/correction terms to increase the pressure and velocity accuracy of reduced order models (ROMs) for fluid flows. Specifically, we propose the first pressure-based data-driven variational multiscale ROM, in which we use the available data to construct closure/correction terms for both the momentum equation and the continuity equation. Our numerical investigation of the two-dimensional flow past a circular cylinder at Re=50000 in the marginally-resolved regime shows that the novel pressure data-driven variational multiscale ROM yields significantly mo…
In this paper, we focus on the mathematical foundations of reduced order model (ROM) closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. Finally, we investigate the verifiability of the data-driven variational multiscale ROM in the numeric…
A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element method for the incompressible Navier-Stokes equations, which combined residual-based stabilization of advection, energetic stability, and satisfaction of a discrete incompressibility condition. However, the convergence analysis and numerical tests of the cited work were subject to the restrictive assumption of a divergence-conforming choice of velocity and …
We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocit…
The proposed method aims to approximate a solution of a fluid-fluid interaction problem in case of low viscosities. The nonlinear interface condition on the joint boundary allows for this problem to be viewed as a simplified version of the atmosphere-ocean coupling. Thus, the proposed method should be viewed as potentially applicable to air-sea coupled flows in turbulent regime. The method consists of two key ingredients. The geometric averaging approach is used for efficient and stable decoupling of the problem, which would allow for the usage of preexisting codes for the air and sea domain s…
A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept
We show that in the variational multiscale framework, the weak enforcement of essential boundary conditions via Nitsche's method corresponds directly to a particular choice of projection operator. The consistency, symmetry and penalty terms of Nitsche's method all originate from the fine-scale closure dictated by the corresponding scale decomposition. As a result of this formalism, we are able to determine the exact fine-scale contributions in Nitsche-type formulations. In the context of the advection-diffusion equation, we develop a residual-based model that incorporates the non-vanishing fin…
A variational multiscale approach to PDE-constrained optimization problems arising in Data-Driven Computational Mechanics
Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE)（国际工程数值方法中心）；Instituto de Ciências Matemáticas e de Computacão, Universidade de São Paulo（圣保罗大学数学与计算科学研究所）；University of Bergen（卑尔根大学）
Three-field Fluid-Structure Interaction (FSI) formulations for fluid and solid are applied and compared to the standard two field-one field formulation for fluid and solid, respectively. Both formulations are applied in a non linear setting for a Newtonian fluid and a neo-Hookean solid in an updated Lagrangian form, both approximated using finite elements and stabilized by means of the Variational Multiscale (VMS) Method to permit the use of arbitrary interpolations. It is shown that this type of coupling leads to a more stable solution. Even though the new formulation poses the necessity of a…
We develop a unified continuum modeling framework for viscous fluids and hyperelastic solids using the Gibbs free energy as the thermodynamic potential. This framework naturally leads to a pressure primitive variable formulation for the continuum body, which is well-behaved in both compressible and incompressible regimes. Our derivation also provides a rational justification of the isochoric-volumetric additive split of free energies in nonlinear continuum mechanics. The variational multiscale analysis is performed for the continuum model to construct a foundation for numerical discretization.…
In this paper, we investigate the hemodynamics of a left atrium (LA) by proposing a computational model suitable to provide physically meaningful fluid dynamics indications and detailed blood flow characterization. In particular, we consider the incompressible Navier-Stokes equations in Arbitrary Lagrangian Eulerian (ALE) formulation to deal with the LA domain under prescribed motion. A Variational Multiscale (VMS) method is adopted to obtain a stable formulation of the Navier-Stokes equations discretized by means of the Finite Element method and to account for turbulence modeling based on Lar…
In this work, we present a computational fluid-structure interaction (FSI) study for a healthy patient-specific pulmonary arterial tree using the unified continuum and variational multiscale (VMS) formulation we previously developed. The unified framework is particularly well-suited for FSI, as the fluid and solid sub-problems are addressed in essentially the same manner and can thus be uniformly integrated in time with the generalized-$α$ method. In addition, the VMS formulation provides a mechanism for large-eddy simulation in the fluid sub-problem and pressure stabilization in the solid sub…
Trajectory-wise data-driven reduced order models (ROMs) tend to be sensitive to training data, and thus lack robustness. We propose to construct a robust stochastic ROM closure (S-ROM) from data consisting of multiple trajectories from random initial conditions. The S-ROM is a low-dimensional time series model for the coefficients of the dominating proper orthogonal decomposition (POD) modes inferred from data. Thus, it achieves reduction both space and time, leading to simulations orders of magnitude faster than the full order model. We show that both the estimated POD modes and parameters in…
A term-by-term variational multiscale method with dynamic subscales for incompressible turbulent aerodynamics
Computational Heat and Fluid Flow Lab, Universidad de Santiago de Chile（圣地亚哥智利大学计算热与流体流动实验室）；Department of Mechanical Engineering, University of Santiago de Chile（圣地亚哥智利大学机械工程系）；Chair for Computational Analysis of Technical Systems, RWTH Aachen University（亚琛工业大学技术系统计算分析教席）；Chair of Methods for Model-based Development in Computational Engineering, RWTH（亚琛工业大学计算工程基于模型开发方法教席）；Center for Simulation and Data Science (JARA-CSD), RWTH Aachen University（亚琛工业大学仿真与数据科学中心（JARA-CSD））
We propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing t…
Beijing Normal-Hong Kong Baptist University（北京岭南师范学院）；Guangzhou Nanfang College（广州南方学院）；Louisiana State University（路易斯安那州立大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
In this study, we present a stabilized finite element analysis for completely unified Stokes-Brinkman problems fully coupled with variable coefficient transient Advection-Diffusion-Reaction equation(VADR). As well we have carried out the stabilized finite element analysis for Stokes-Brinkman model with interface conditions fully coupled with VADR. The viscosity of the fluid, involved in flow problem, depends on the concentration of the solute, whose transport is described by VADR equation. The algebraic subgrid multiscale approach has been employed to arrive at the stabilized coupled variation…
In this paper a fully coupled system of transient $Navier$-$Stokes$ ($NS$) fluid flow model and variable coefficient unsteady Advection-Diffusion-Reaction ($VADR$) transport model has been studied through subgrid multiscale stabilized finite element method. In particular algebraic approach of approximating the subscales has been considered to arrive at stabilized variational formulation of the coupled system. This system is strongly coupled since viscosity of the fluid depends upon the concentration, whose transportation is modelled by $VADR$ equation. Fully implicit schemes have been consider…
The simulation of the elastodynamics equations at high-frequency suffers from the well known pollution effect. We present a Petrov--Galerkin multiscale sub-grid correction method that remains pollution free in natural resolution and oversampling regimes. This is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the $\log(k)$, $k$ being the wavenumber. Key to this method are polynomial-in-$k$ bounds for stability constants and related inf-sup constants. To this end, we establish polynomial-in-$k$ bounds for the elastodynamic…
Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation with Time Oscillation
National Center for Applied Mathematics（应用数学国家研究中心）；Tianjin University（天津大学）；School of Mechanical and Aerospace Engineering, Jilin University（吉林大学机械与航空航天工程学院）；Jilin University（吉林大学）
