Variational Multiscale Research Articles

A posteriori error estimates in a finite element VMS-based reduced order model for the incompressible Navier-Stokes equations

In this paper we present an a posteriori error estimate for a reduced order model (ROM) for the incompressible Navier-Stokes equations that is based on the fact that the full order model is a finite element (FE) approximation. Both this FE approximation and the ROM are stabilized by means of a variational multi-scale (VMS) strategy, in which the unknowns are split into FE scales and sub-grid scales (SGS), the latter being modeled in terms of the former. The SGS, when properly scaled, provide directly the a posteriori error estimate, both for the ROM and for the FE approximation.

Primal interface debonding formulation for finite strain isotropic plasticity

A framework is developed for modeling ductile damage of nonlinear materials whose plastic deformation is characterized using rate independent classical plasticity. This method relies on the assumption that the free energy can be decomposed into elastic, plastic and damage parts. A thermodynamically consistent method is derived which satisfies the second law of thermodynamics in the Clausius–Duhem inequality form. The dissipation associated with plasticity takes place in the domain only, while damage dissipation is localized to the interface. The method is developed using Variational Multiscale ideas to obtain definitions of the interface fluxes within a primal formulation analogous to the Discontinuous Galerkin method, which ensures weakly vanishing interface gap prior to reaching a damage initiation criterion. The local nonlinear problem to calculate both plastic deformation gradient and damage variable follows an incremental approach similar to classical plasticity return mapping algorithm. This elastoplastic damage formulation is developed for material undergoing finite strain, and it naturally accommodates a trapezoidal traction separation law (TSL) whose shape can be varied to model either ductile interface behavior or brittle interface behavior. The formulation's performance is assessed through modeling a patch test and a compact tension specimen.

An adaptive Finite Element strategy for the numerical simulation of additive manufacturing processes

In this work an adaptive Finite Element strategy to deal with the numerical simulation of Additive Manufacturing (AM) processes is presented. The Selective Laser Melting (SLM) is chosen as the reference technology because of its great diffusion in the industrial manufacturing chain, although the proposed methodology can be applied to the numerical simulation of all types of AM. An octree-based mesh adaptivity approach is adopted allowing for the use of much finer meshes within the processing zone, the so called Thermo-Mechanically Affected Zone (TMAZ), if compared to the rest of the computational domain. Although the adaptive meshing is vital to keep controlled the computational resources through the entire simulation of the fabrication process, the accuracy of the results can be compromised by the coarsening strategy, and particularly when simulating the SLM process, where the mesh size can vary from microns (TMAZ) to centimetres (close to the build-plate). This loss of accuracy can spoil the original efforts in refining the mesh in the process zone. Therefore a strategy to compensate for information loss in the adaptive refinement simulation of additive manufacturing processes is developed. The main idea is to add two correction terms which compensate for the loss of accuracy in the coarsening process of the mesh in the already manufactured regions. The proposed correction terms can be interpreted as a Variational Multiscale enhancement on the adaptive mesh. This allows one to successfully simulate the additive manufacturing process by using an adaptively coarsened mesh, with results which have an accuracy very similar to the one of a uniformly refined mesh simulation, at a fraction of the computational cost. Numerical examples illustrate the performance of the proposed strategy.

Data-driven variational multiscale reduced order models

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 the interactions among the three types of scales. In the third step, we use available data to model the VMS–ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS–ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS–ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient ν=10−3; (ii) a 2D flow past a circular cylinder at Reynolds numbers Re=100, Re=500, and Re=1000; (iii) the quasi-geostrophic equations at Reynolds number Re=450 and Rossby number Ro=0.0036; and (iv) a 2D flow over a backward facing step at Reynolds number Re=1000. The numerical results show that the data-driven VMS–ROM is significantly more accurate than standard ROMs.

A Variational Multiscale method with immersed boundary conditions for incompressible flows.

This paper presents a new stabilized form of incompressible Navier-Stokes equations for weak enforcement of Dirichlet boundary conditions at immersed boundaries. The boundary terms are derived via the Variational Multiscale (VMS) method which involves solving the fine-scale variational problem locally within a narrow band along the boundary. The fine-scale model is then variationally embedded into the coarse-scale form that yields a stabilized method which is free of user defined parameters. The derived boundary terms weakly enforce the Dirichlet boundary conditions along the immersed boundaries that may not align with the inter-element edges in the mesh. A unique feature of this rigorous derivation is that the structure of the stabilization tensor which emerges is naturally endowed with the mathematical attributes of area-averaging and stress-averaging. The method is implemented using 4-node quadrilateral and 8-node hexahedral elements. A set of 2D and 3D benchmark problems is presented that investigate the mathematical attributes of the method. These test cases show that the proposed method is mathematically robust as well as computationally stable and accurate for modeling boundary layers around immersed objects in the fluid domain.

Space\u2013Time Variational Multiscale Isogeometric Analysis of a tsunami-shelter vertical-axis wind turbine

We present computational flow analysis of a vertical-axis wind turbine (VAWT) that has been proposed to also serve as a tsunami shelter. In addition to the three-blade rotor, the turbine has four support columns at the periphery. The columns support the turbine rotor and the shelter. Computational challenges encountered in flow analysis of wind turbines in general include accurate representation of the turbine geometry, multiscale unsteady flow, and moving-boundary flow associated with the rotor motion. The tsunami-shelter VAWT, because of its rather high geometric complexity, poses the additional challenge of reaching high accuracy in turbine-geometry representation and flow solution when the geometry is so complex. We address the challenges with a space–time (ST) computational method that integrates three special ST methods around the core, ST Variational Multiscale (ST-VMS) method, and mesh generation and improvement methods. The three special methods are the ST Slip Interface (ST-SI) method, ST Isogeometric Analysis (ST-IGA), and the ST/NURBS Mesh Update Method (STNMUM). The ST-discretization feature of the integrated method provides higher-order accuracy compared to standard discretization methods. The VMS feature addresses the computational challenges associated with the multiscale nature of the unsteady flow. The moving-mesh feature of the ST framework enables high-resolution computation near the blades. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA enables more accurate representation of the blade and other turbine geometries and increased accuracy in the flow solution. The STNMUM enables exact representation of the mesh rotation. A general-purpose NURBS mesh generation method makes it easier to deal with the complex turbine geometry. The quality of the mesh generated with this method is improved with a mesh relaxation method based on fiber-reinforced hyperelasticity and optimized zero-stress state. We present computations for the 2D and 3D cases. The computations show the effectiveness of our ST and mesh generation and relaxation methods in flow analysis of the tsunami-shelter VAWT.

Computational Flow Analysis in Aerospace, Energy and Transportation Technologies with the Variational Multiscale Methods

With the recent advances in the variational multiscale (VMS) methods, computational ow analysis in aerospace, energy, and transportation technologies has reached a high level of sophistication. It is bringing solutions in challenging problems such as the aerodynamics of parachutes, thermo-fluid analysis of ground vehicles and tires, and fluid-structure interaction (FSI) analysis of wind turbines. The computational challenges include complex geometries, moving boundaries and interfaces, FSI, turbulent flows, rotational flows, and large problem sizes. The Residual-Based VMS (RBVMS), Arbitrary Lagrangian-Eulerian VMS (ALE-VMS) and Space-Time VMS (ST-VMS) methods have been successfully serving as core methods in addressing the computational challenges. The core methods are supplemented with special methods targeting specific classes of problems, such as the Slip Interface (SI) method, MultiDomain Method, and the ST-C data compression method. We provide and overview of the core and special methods. We present, as examples of challenging computations performed with these methods, aerodynamic analysis of a ramair parachute, thermo-fluid analysis of a freight truck and its rear set of tires, and aerodynamic and FSI analysis of two back-to-back wind turbines in atmospheric boundary layer flow. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Fluid-structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation

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-problem. The FSI problem is solved in a quasi-direct approach, in which the pressure and velocity in the unified continuum body are first solved, and the solid displacement is then obtained via a segregated algorithm and prescribed as a boundary condition for the mesh motion. Results of the pulmonary arterial FSI simulation are presented and compared against those of a rigid wall simulation.

Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes

We report on simulations of two-phase flows with deforming interfaces at various density contrasts by solving thermodynamically consistent Cahn-Hilliard Navier-Stokes equations. An (essentially) unconditionally energy-stable Crank-Nicolson-type time integration scheme is used. Detailed proofs of energy stability of the semi-discrete scheme and for the existence of solutions of the advective-diffusive Cahn-Hilliard operator are provided. We discretize spatial terms with a conforming continuous Galerkin finite element method in conjunction with a residual-based variational multi-scale (VMS) approach in order to provide pressure stabilization. We deploy this approach on a massively parallel numerical implementation using fast octree-based adaptive meshes. A detailed scaling analysis of the solver is presented. Numerical experiments showing convergence and validation with experimental results from the literature are presented for a large range of density ratios.

Evaluation of a nonlinear variational multiscale method for fluid transport problems

Diverse transport problems, especially those based on fluid flow models, are intrinsically multiscale and nonlinear, characteristics that often lead to intricate dynamics such as the development of instabilities and turbulence. Computational simulations that resolve all scales in these problems are often unfeasible, prompting to coarse-grained simulation strategies in which small-scale features are modeled instead of resolved. Variational Multiscale (VMS) methods, and particularly residual-based Large-Eddy Simulation (LES) approaches, have proven effective and robust for the coarse-grained simulation of complex transport problems. VMS methods avoid the assumption of separable nonlinearity and the reliance on empirical small-scale models by using a variational decomposition of scales together with a residual-based approximation of the small-scales. Evaluation of a nonlinear VMS approach, denoted as VMSn, is presented for the coarse-grained simulation of transient-advective-diffusive-reactive (TADR) transport problems arising from fluid flow models. In contrast to classical VMS approaches that neglect the effect of the small scales on the transport operator, VMSn treats the inter-dependence between large- and small-scales upfront. The treatment of inter-scale coupling involves the solution of a local algebraic nonlinear system describing the evolution of the small-scales. The VMSn approach is complemented with two algebraic approximations of the small-scales: one based on the main diagonal of the transport matrices and another that preserves transport fluxes and is suitable for generic TADR systems. The suitability of the VMSn approach for handling general TADR problems and regimes is evaluated with benchmark incompressible, compressible, and magnetohydrodynamic laminar flow problems, the incompressible Taylor-Green vortex flow, the turbulent free jet, and the two-temperature arc in crossflow. Simulation results show that VMSn leads to minor improvements in accuracy with respect to the classical VMS for the laminar flow problems, but to significantly greater accuracy for the turbulent flows and the unsteady plasma flow problems, while using the same cohesive numerical formulation.

A variational multiscale framework for atmospheric turbulent flows over complex environmental terrains

A residual-based variational multi-scale (VMS) modeling framework is applied to simulate atmospheric flow over complex environmental terrains. The VMS framework is verified and validated using two test cases using linear finite element (FEM) and quadratic non-uniform rational B-spline (NURBS) discretizations. First, the flow over the 3D, axisymmetric Gaussian hill (normally distributed surface) is used to compare the results with FEM and NURBS discretization. Next, the actual terrain of the Bolund hill is used to demonstrate the efficacy of the framework. Good agreement with published data, coming from simulations and field measurements, is achieved, with the NURBS discretization showing much better per-degree-of-freedom accuracy compared with linear FEM. The paper includes a comprehensive review of experimental and numerical methods, and the corresponding challenges, for complex-terrain flows, which provides a proper context for the developments presented in this work.

Variational multiscale closures for finite element discretizations using the Mori–Zwanzig approach

The simulation of multiscale problems remains a challenge in many problems due to the complex interaction between the resolved and unresolved scales. This work develops a coarse-grained modeling approach for Galerkin discretizations by combining the Variational Multiscale decomposition and the Mori–Zwanzig (M–Z) formalism. An appeal of the M–Z formalism is that – akin to Green’s functions for linear problems – the impact of unresolved dynamics on resolved scales can be formally represented as a convolution (or memory) integral in a non-linear setting. To ensure tractable and efficient models, Markovian closures are developed for the M–Z memory integral. The resulting sub-scale model has some similarities to adjoint stabilization and orthogonal sub-scale models. The model is made parameter-free by adaptively determining the memory length during the simulation. To illustrate the generalizability of this model, the performance is assessed in detail in coarse-grained simulations of a range of problems from the one-dimensional Burgers equation and to incompressible turbulence.

Solution of transient viscoelastic flow problems approximated by a term-by-term VMS stabilized finite element formulation using time-dependent subgrid-scales

Some finite element stabilized formulations for transient viscoelastic flow problems are presented in this paper. These are based on the Variational Multiscale (VMS) method, following the approach introduced in Castillo and Codina et al. (2019), for the Navier–Stokes problem, the main feature of the method being that the time derivative term in the subgrid-scales is not neglected. The main advantage of considering time-dependent sub-grid scales is that stable solutions for anisotropic space–time discretizations are obtained; however other benefits related with elastic problems are found along this study. Additionally, a split term-by-term stabilization method is discussed and redesigned, where only the momentum equation is approached using a term-by-term methodology, and which turns out to be much more efficient than other residual-based formulations. The proposed methods are designed for the standard and logarithmic formulations in order to deal with high Weissenberg number problems in addition to anisotropic space–time discretizations, ensuring stability in all cases. The proposed formulations are validated in several benchmarks such as the flow over a cylinder problem and the lid-driven cavity problem, obtaining stable and accurate results. A comparison between formulations and stabilization techniques is done to demonstrate the efficiency of time-dependent sub-grid scales and the term-by-term methodologies.

Variational multiscale modeling with discretely divergence-free subscales

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 H1-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 velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier–Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.

An assessment of two classes of variational multiscale methods for the simulation of incompressible turbulent flows

A numerical assessment of two classes of variational multiscale (VMS) methods for the simulation of incompressible flows is presented. Two types of residual-based VMS methods and two types of projection-based VMS methods are included in this assessment. The numerical simulations are performed at turbulent channel flow problems with various friction Reynolds numbers. It turns out the residual-based VMS methods, in particular when used with a pair of inf–sup stable finite elements, give usually the most accurate results for second order statistics. For this pair of finite element spaces, a flexible GMRES method with a Least Squares Commutator (LSC) preconditioner proved to be an efficient solver.

Wind Turbine and Turbomachinery Computational Analysis with the ALE and Space-Time Variational Multiscale Methods and Isogeometric Discretization

The challenges encountered in computational analysis of wind turbines and turbomachinery include turbulent rotational flows, complex geometries, moving boundaries and interfaces, such as the rotor motion, and the fluid-structure interaction (FSI), such as the FSI between the wind turbine blade and the air. The Arbitrary Lagrangian-Eulerian (ALE) and Space-Time (ST) Variational Multiscale (VMS) methods and isogeometric discretization have been effective in addressing these challenges. The ALE-VMS and ST-VMS serve as core computational methods. They are supplemented with special methods like the Slip Interface (SI) method and ST Isogeometric Analysis with NURBS basis functions in time. We describe the core and special methods and present, as examples of challenging computations performed, computational analysis of horizontal and vertical-axis wind turbines and flow-driven This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Computational analysis of particle-laden-airflow erosion and experimental verification

Computational analysis of particle-laden-airflow erosion can help engineers have a better understanding of the erosion process, maintenance and protection of turbomachinery components. We present an integrated method for this class of computational analysis. The main components of the method are the residual-based Variational Multiscale (VMS) method, a finite element particle-cloud tracking (PCT) method with ellipsoidal clouds, an erosion model based on two time scales, and the Solid-Extension Mesh Moving Technique (SEMMT). The turbulent-flow nature of the analysis is addressed with the VMS, the particle-cloud trajectories are calculated based on the time-averaged computed flow field and closure models defined for the turbulent dispersion of particles, and one-way dependence is assumed between the flow and particle dynamics. Because the target-geometry update due to the erosion has a very long time scale compared to the fluid–particle dynamics, the update takes place in a sequence of “evolution steps” representing the impact of the erosion. A scale-up factor, calculated based on the update threshold criterion, relates the erosions and particle counts in the evolution steps to those in the PCT computation. As the target geometry evolves, the mesh is updated with the SEMMT. We present a computation designed to match the sand-erosion experiment we conducted with an aluminum-alloy target. We show that, despite the problem complexities and model assumptions involved, we have a reasonably good agreement between the computed and experimental data.

Variational multiscale framework for cavitating flows

A numerical formulation for the modeling of turbulent cavitating flows is presented. The flow field is governed by the 3D, time-dependent Navier–Stokes equations for a compressible isothermal mixture. The Arbitrary Lagrangian–Eulerian Variational Multiscale (ALE-VMS) formulation is adopted to model the turbulent flow on moving domains with no-slip boundary conditions imposed weakly. The formulation is first tested on the cavitating flow over a 2D NACA0012 airfoil and compared to published numerical results. Next, the framework is applied to the benchmark problem for the flow over a hemispherical fore-body. The numerical results are compared to the reported experimental data, showing a good agreement over the range of cavitation numbers. Finally, the simulation of a hydrokinetic turbine in cavitating flow at a low cavitation number is presented in order to test the stability of the formulation and the capability to handle real engineering problems involving turbulent cavitating flows on moving domains.

Interfacial coupling across incompatible meshes in a monolithic finite-strain thermomechanical formulation

A stabilized interface method is presented for monolithic coupling of thermomechanical fields across incompatible internal interfaces. Embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework yields interface coupling terms that accommodate nonmatching discretizations across glued meshes. A unique feature of the method that emanates from fine-scale modeling facilitated by VMS is the derivation of analytical expressions for the Lagrange multipliers that enforce continuity of the fields across nonmatching meshes. This stabilized form is free of any user-defined parameters and renders itself to consistent linearization that leads to quadratic convergence of the method. The coupling terms at the interface also provide an avenue to employ various kinematic models for interfacial response in multi-constituent materials. An interfacial gap/crack function is introduced together with the conjugate traction vector to model discrete interfacial failure in multi-material applications. An algorithm for consistent implementation of interfacial models for tension, compression, and sliding friction across non-conforming meshes is presented. Test cases that employ 8-node hexahedra and 4-node tetrahedra are presented that highlight the generality of the method and validate its robustness for mathematically nonsmooth thermomechanical problems.

Fluid structure interaction by means of variational multiscale reduced order models

SummaryA reduced order model designed by means of a variational multiscale stabilized formulation has been applied successfully to fluid‐structure interaction problems in a strongly coupled partitioned solution scheme. Details of the formulation and the implementation both for the interaction problem and for the reduced models, for both the off‐line and on‐line phases, are shown. Results are obtained for cases in which both domains are reduced at the same time. Numerical results are presented for a semistationary and a fully transient case.