Variational Multiscale Method Research Articles

Adaptive Immersed Mesh Method (AIMM) for Fluid–Structure Interaction

Our paper proposes an innovative approach for modeling Fluid-Structure Interaction (FSI).Our method combines both traditional monolithic and partitioned approaches, creating a hybrid solution that facilitates FSI. At each time iteration, the solid mesh is immersed within a fluid-solid mesh, all while maintaining its independent Lagrangian hyperelastic solver. The Eulerian mesh encompasses both the fluid and solid components and accommodates various physical phenomena. We enhance the interaction between solid and fluid through anisotropic mesh adaptation and the Level-Set methods. This enables a more accurate representation of their interaction. Together, these components constitute the Adaptive Immersed Mesh Method (AIMM). For both solvers, we utilize the Variational Multi-Scale (VMS) method, mitigating potential spurious oscillations common with piecewise linear tetrahedral elements. The framework operates in 3D with parallel computing capabilities. Our method’s accuracy, robustness, and capabilities are assessed through a series of 2D numerical problems. Furthermore, we present various three-dimensional test cases and compare their results to experimental data.

Multi-Objective Topology Optimization of Conjugate Heat Transfer Using Level Sets and Anisotropic Mesh Adaptation

This study proposes a new computational framework for the multi-objective topology optimization of conjugate heat transfer systems using a continuous adjoint approach. It relies on a monolithic solver for the coupled steady-state Navier–Stokes and heat equations, which combines finite elements stabilized by the variational multi-scale method, level set representations of the fluid–solid interfaces and immersed modeling of heterogeneous materials (fluid–solid) to ensure that the proper amount of heat is exchanged to the ambient fluid by solid objects in arbitrary geometry. At each optimization iteration, anisotropic mesh adaptation is applied in near-wall regions automatically captured by the level set. This considerably cuts the computational effort associated with calling the finite element solver, in comparison to traditional topology optimization algorithms operating on isotropic grids with a comparable refinement level. Given that we operate within the constraint of a specified number of nodes in the mesh, this allows not only to improve the accuracy of interface representation and motion but also to retain the high fidelity of the numerical solutions at the grid points just adjacent to the interface. Finally, the remeshing and resolution steps both run within a highly parallel environment, which makes it possible for the proposed algorithm to tackle large-scale problems in three dimensions with several tens of millions of state degrees of freedom. The developed solver is validated first by minimizing dissipation in a flow splitter device, for which the method delivers relevant optimal designs over a wide range of volume constraints and flow rate distributions over the multiple outlet orifices but yields better accuracy compared to reference data from literature obtained using uniform meshes (in the sense that the layouts are more smooth, and the solutions are better resolved). The scheme is then applied to a two-dimensional heat transfer problem, using bi-objective cost functionals combining flow resistance and thermal recoverable power. A comprehensive parametric study reveals a complex arrangement of optimal solutions on the Pareto front, with multiple branches of symmetric and asymmetric designs, some of them previously unreported. Finally, the algorithmic developments are substantiated with several three-dimensional numerical examples tackled under fixed weights for heat transfer and flow resistance, for which we show that the optimal layouts computed at low Reynolds number, that are intrinsically relevant to a broad range of microfluidic application, can also serve as smooth solutions to high-Reynolds-number engineering problems of practical interest.

GO-MELT: GPU-optimized multilevel execution of LPBF thermal simulations

Computationally modeling the laser powder bed fusion process can be expensive despite the fact that the laser covers a small fraction of the entire domain. Hence, we developed GO-MELT, a multilevel approach inspired by the variational multiscale method. GO-MELT calculates the full temperature solution using three coupled thermal solvers, each solving one of three overlapping domains, subsequently referred to as levels. Level 1’s solver obtains a global, coarse-scale solution. Level 2’s solver computes a meso-scale thermal field spanning the melt pool region. Level 3’s solver obtains a fine-scale temperature field spanning the laser’s immediate region. Coarser solutions provide boundary conditions for finer solvers. Finer solutions compute subgrid scales that influence coarser solvers through additional source terms. Being independently meshed, Levels 2 and 3 can track a laser along its tool path without remeshing. Moreover, fixed-sized, structured meshes allow for GPU acceleration using Google’s JAX library with just-in-time compilation. Five case studies were conducted to demonstrate proper convergence of GO-MELT, to show that GO-MELT can reach the same accuracy as a uniform mesh with fewer degrees of freedom, and to quantify the computational load from each level’s solver. A simulated production run averaged 1.64ms per time step after taking 9.36h to complete 20.5 million time steps, which is approximately 678× faster than a uniform mesh solver with identical resolution and GPU acceleration. Future work will implement different solvers into GO-MELT to improve fidelity and speed.

High-resolution 3D computation of time-periodic long-wake flows with the Carrier-Domain Method and Space–Time Variational Multiscale method with isogeometric discretization

High-resolution 3D computation of time-periodic long-wake flows with the Carrier-Domain Method and Space–Time Variational Multiscale method with isogeometric discretization

Partitioned schemes for the blood solute dynamics model by the variational multiscale method

In this paper, we consider a heterogeneous model of solute absorption processes by the arterial wall. This model is based on an advection-diffusion equation describing the solute dynamics in the vascular lumen, the convective field being provided by the blood velocity. A pure diffusive model coupled with this equation is considered for the solute dynamics inside the arterial wall. The two subdomains are physically separated by the endothelial layer, which acts as a selectively permeable membrane and the interface condition matching the two subproblems follow from the nature of this membrane. To compute the approximate solutions, we propose two partitioned schemes for this model by the variational multiscale method. Stability and convergence results are proved for both schemes. We derive error bounds of the fully discrete solution which are first order in time. The optimal error estimates in space could be achieved for the velocity and concentration in the H1-norm, and pressure in the L2-norm with the proper choosing of stabilized parameters. Theoretical results are supported by numerical examples, and two model problems from the physiological interest are also considered.

A stabilized finite element method for modeling dispersed multiphase flows using orthogonal subgrid scales

We propose a finite-element formulation for simulating multi-component flows occupying the same domain with spatially varying concentrations. Each constituent is assumed to behave as an incompressible Newtonian fluid, and solutions are sought for the velocities and volume fractions of each phase, as well as the common pressure. Stabilization terms are derived within the framework of the variational multiscale method based on an approximation of the finite-element residual to achieve control of the pressure and volume fractions. We utilize the concept of term-by-term stabilization in conjunction with orthogonal subgrid scales, thus incorporating only those terms of the residual essential to obtain stability and projecting them on a space orthogonal to the finite element space. The resulting system of equations is solved in a monolithic manner, requiring a small number of nonlinear iterations. Several benchmark tests have been performed to confirm the stability and optimal asymptotic convergence rates for linear and higher-order elements using the proposed formulation.

Accounting for mesoscale geometry and intra-yarn fiber volume fraction distribution on 3D angle-interlock fabric permeability

Understanding resin flows within 3D interlock fabrics involves addressing a dual-scale flow problem. Indeed, the fluid can flow through inter-yarn channels, characterized by a unit cell mesoscale morphology, but also within yarns considered as homogeneous equivalent porous media. The former is assumed to follow the Stokes’ law while the latter be related to the Darcy’s law, directly linked to the microscopic intra-yarn fiber volume fraction (FVF) field. In this work, a coupled Stokes–Darcy steady-state flow within a 3D woven textile, modeled at mesoscale, is solved through a finite element monolithic approach with a mixed velocity-pressure formulation, stabilized by the Variational MultiScale Method (VMS) and implemented in the Z-set software. The fabric permeability tensor is then computed without any assumptions on its nature and analyzed both through its diagonal components, its eigenvectors and eigenvalues. The main contribution of this study lies in examining the influence of variations of the inter-yarn porous medium morphology, through textile compaction and geometrical reduction to prevent edge flows, and the intra-yarn permeability (from complete field to unique value) on the overall fabric permeability.

Stabilized mixed material point method for incompressible fluid flow analysis

This paper proposes novel and robust stabilization strategies for accurately modeling incompressible fluid flow problems in the material point method (MPM). To address the modeling of Newtonian fluids with incompressibility constraints, a new mixed implicit MPM formulation is proposed. Here, instead of solving the velocity and pressure fields as the unknown variables like the typical Eulerian computational fluid dynamics (CFD) solver, the proposed approach adopts a monolithic displacement–pressure formulation inspired by the mixed-form updated-Lagrangian Finite Element Method (FEM). To satisfy the inf–sup stability condition, two stabilization strategies are integrated into the formulation: the variational multiscale method (VMS) and the pressure-stabilization Petrov–Galerkin method (PSPG). By concurrently solving the displacement and pressure fields, the developed monolithic solver obviates the need for free-surface detection as well as Dirichlet and Neumann pressure imposition, in contrast to the fractional-step method. This attribute mitigates spurious pressure and velocity oscillations in simulating dynamic and transient flow problems. This study also addresses other prevalent challenges in MPM simulations, such as the pressure oscillations triggered by cell-crossing errors, particle-distribution-induced quadrature errors, and particle-grid information transfer. To resolve these issues, the quadratic B-Spline basis function, the delta-correction method, and the Taylor particle-in-cell method are incorporated into the proposed mixed MPM formulation, thereby enhancing numerical stability. The efficacy of the proposed stabilized incompressible MPM framework is validated through several benchmark cases, comparing the obtained results with other numerical methods and analytical solutions. Furthermore, the method’s capability in simulating real-world problems involving violent free-surface fluid motion is demonstrated through comparisons with experimental results of water sloshing and dam break scenarios.

An efficient variational multiscale element free Galerkin method based on adaptive TR-AB2 scheme for convection–diffusion equations

The variational multiscale element-free Galerkin (VMEFG) method can effectively improve stability when solving convection-dominated problems. However, for time-dependent problems, the VMEFG method has a large amount of calculation. In this paper, an efficient VMEFG method for solving unsteady convection-dominated convection–diffusion equations is proposed. The trapezoidal rule (TR) with adaptive time step is used for time integration, and the explicit second-order Adams–Bashforth (AB2) method is used to control the error. The main idea is to allow the time step to change along the direction of the solution, so that the amount of calculation under the specified accuracy requirements can be minimized. We further exemplified the effectiveness and accuracy of the given method in several examples. The numerical results show that the VMEFG method combined with TR-AB2 method (VMEFG-TR-AB2) can effectively increase computing efficiency while maintaining the calculation accuracy for the convection-dominated problem.

Stabilized finite elements for the solution of the Reynolds equation considering cavitation

The Reynolds equation, combined with the Elrod algorithm for including the effect of cavitation, resembles a nonlinear convection–diffusion–reaction (CDR) equation. Its solution by finite elements is prone to oscillations in convection-dominated regions, which are present whenever cavitation occurs. We propose a stabilized finite-element method that is based on the variational multiscale method and exploits the concept of orthogonal subgrid scales. We demonstrate that this approach only requires one additional term in the weak form to obtain a stable method that converges optimally when performing mesh refinement.

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

Modeling of multi-phase, multi-fluid flows with applications to marine hydrokinetic turbines

In this paper, a numerical formulation for modeling turbulent multi-phase, multi-fluid flows over real-life marine devices is presented. The flow field is governed by the Navier–Stokes equations with an Arbitrary-Lagrangian–Eulerian (ALE) description of the continuum for handling independent domain movements. The multi-phase phenomenon is modeled for an iso-thermal homogeneous mixture of water vapor and liquid phases using a transport equation for the vapor volume fraction. The multi-fluid phenomenon is modeled using the level-set method where a geometric function, namely, the signed distance function (SDF) describes the level-set field. Turbulence modeling is handled in the context of the variational multiscale (VMS) method resulting in an ALE-VMS formulation for multi-phase, multi-fluid flows over moving hydrodynamic objects. Several augmentations to the formulation are discussed in this paper including the sliding interface (SI) operator and its role in enforcing the compatibility of solution fields between in-contact computational subdomains during the different steps of the solution. This formulation is validated against experimental data for a real-life horizontal-axis marine turbine under different flow conditions including single-phase flows, multi-phase flows and multi-fluid flows. The performance of the turbine as well as the expected flow behaviors are compared to the experimental data with success. An additional case is simulated for a multi-phase, multi-fluid flow condition showing the capabilities and robustness of the presented numerical formulation.

Hyperreduced-order modeling of thermally coupled flows

This article presents a full and reduced-order methodology based on the finite element method that allows the characterization of convective-dominant conjugate heat transfer flows. Reduced-order modeling (ROM) allows the description of several degrees of freedom problems, referred to as full-order modeling (FOM), through a reduced-order surrogate representation based on a Petrov-Galerkin projection onto a reduced subspace. The FOM and ROM problems are stabilized through the Variational Multi-Scale (VMS) method, which ensures stability in dominant convective problems and allows equal interpolation spaces for the pressure, velocity, and temperature. The proposed methodology is validated solving natural heat convection in a differentially heated square cavity, considering air as the working fluid and Rayleigh numbers in the interval 104≤Ra≤108. In this problem, the benefit of using a ROM approximation for the three unknown fields, i.e., velocity, pressure, and temperature, is compared against ROM approximation only in the velocity and pressure fields and a FOM type approximation for temperature. The benefit of using mesh-based hyperreduction and its computational performance and precision are also analyzed. Subsequently, the method simulates a conjugate heat transfer problem that includes natural heat convection, allowing the inclusion of different rheologies between the working fluids. The results verify the precision and speed up of the calculations by more than three hundred times with respect to the time necessary to solve a full-order formulation, emerging it as a great potential tool in the resolution of thermally coupled flows.

Space–time flow computation with boundary layer and contact representation: a 10-year history

In computation of flow problems with moving solid surfaces, moving-mesh methods such as the space–time (ST) variational multiscale method enable mesh-resolution control near the solid surfaces and thus high-resolution boundary-layer representation. There was, however, a perception that in computations where the solid surfaces come into contact, high-resolution boundary-layer representation and actual-contact representation without leaving a mesh protection opening between the solid surfaces were mutually exclusive objectives in a practical sense. The introduction of the ST topology change (ST-TC) method in 2013 changed the perception. The two objectives were no longer mutually exclusive. The ST-TC makes moving-mesh computation possible even without leaving a mesh protection opening. The contact is represented as an actual contact and the boundary layer is represented with high resolution. Elements collapse or are reborn as needed, and that is attainable in the ST framework while retaining the computational efficiency at a practical level. The ST-TC now has a 10-year history of achieving the two objectives that were long seen as mutually exclusive. With the ST-TC and other ST computational methods introduced before and after, it has been possible to address many of the challenges encountered in conducting flow analysis with boundary layer and contact representation, in the presence of additional intricacies such as geometric complexity, isogeometric discretization, and rotation or deformation of the solid surfaces. The flow analyses conducted with these ST methods include car and tire aerodynamics with road contact and tire deformation and ventricle-valve-aorta flow. To help widen awareness of these methods and what they can do, we provide an overview of the methods, including those formulated in the context of isogeometric analysis, and the computations performed over the 10-year history of the ST-TC.

On a fully-implicit VMS-stabilized FE formulation for low Mach number compressible resistive MHD with application to MCF

This study presents the development and evaluation of a fully-implicit variational multiscale (VMS) stabilized unstructured finite element (FE) formulation for compressible magnetohydrodynamics (MHD) model, at low Mach number regime. The model describes the dynamics of a compressible conducting fluid in the low Mach number limit in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics phenomena. The specific applications that motivate this study are macroscopic simulations of the longer time-scale stability and disruptions of magnetic confinement fusion (MCF) devices, specifically the ITER tokamak. The discussion considers the development of the VMS FE representation, the structure of the stabilizing terms that deal with significant convective flows, the stabilization of the nearly incompressible response of the fluid flow, and the stabilization of the constraint that enforces the solenoidal involution on the magnetic field. The nonlinear discretized system is solved with scalable preconditioned Newton–Krylov iterative methods, which employs a multiphysics block preconditioning method based on approximate block factorizations and Schur complements. The study presents an evaluation of the VMS method on a 2D cartesian tearing mode instability, and illustrates the scalability of the solvers on MCF relevant problems. A set of results are also presented for longer time-scale stability and disruptions for the ITER tokamak. These include a vertical displacement event (VDE), and a (1,1) internal kink mode. The formulation is demonstrated to be scalable and also reasonably robust with respect to the Lundquist number scaling.

A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms

This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework’s performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem).

Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction

AbstractIn this article, a methodology for fine scale modeling of large scale linear elastic structures is proposed, which combines the variational multiscale method, domain decomposition and model order reduction. The influence of the fine scale on the coarse scale is modeled by the use of an additive split of the displacement field, addressing applications without a clear scale separation. Local reduced spaces are constructed by solving an oversampling problem with random boundary conditions. Herein, we inform the boundary conditions by a global reduced problem and compare our approach using physically meaningful correlated samples with existing approaches using uncorrelated samples. The local spaces are designed such that the local contribution of each subdomain can be coupled in a conforming way, which also preserves the sparsity pattern of standard finite element assembly procedures. Several numerical experiments show the accuracy and efficiency of the method, as well as its potential to reduce the size of the local spaces and the number of training samples compared to the uncorrelated sampling.

Space–time computational flow analysis: Unconventional methods and first-ever solutions

The Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method was introduced in 1990 as a moving-mesh method for computational analysis of flows with moving boundaries and interfaces (MBI), which is a wide class of problems that includes fluid–particle and fluid–structure interactions and free-surface and multi-fluid flows. The method was inspired by Thomas Hughes’s 1987 work on space–time finite element methods for elastodynamics. The original DSD/SST method is now called “ST-SUPS”, reflecting its stabilization components, which are the Streamline-Upwind/Petrov–Galerkin (SUPG) method, pioneered by Hughes, and the Pressure-Stabilizing/Petrov–Galerkin (PSPG) method, inspired by Hughes’s work on a Stokes-flow Petrov–Galerkin formulation allowing equal-order interpolations for velocity and pressure. Hughes’s work on the residual-based variational multiscale (RBVMS) method inspired the ST-VMS method, which is the VMS version of the DSD/SST. A number of special methods were introduced in connection with the core methods ST-SUPS and ST-VMS. Hughes’s work on isogeometric analysis (IGA) inspired one of those special methods, the “ST-IGA”, with IGA basis functions not only in space but also in time. The core and special ST methods enabled first-ever solutions in some of the most challenging classes of MBI problems, including particle-laden flows, spacecraft parachute fluid–structure interactions, and car and tire aerodynamics. We provide an overview of the ST methods inspired by Hughes’s work and highlight some of the first-ever solutions in these three classes of MBI problems.

Variational multiscale method stabilization parameter calculated from the strain-rate tensor

The stabilization parameters of the methods like the Streamline-Upwind/Petrov–Galerkin, Pressure-Stabilizing/Petrov–Galerkin, and the Variational Multiscale method typically involve two local length scales. They are the advection and diffusion length scales, appearing in the expressions for the advective and diffusive limits of the stabilization parameter. The advection length scale has always been in the flow direction. The diffusion length scales in use have mostly been just element-geometry-dependent, but there is good justification for also making that direction-dependent, so that the spatial variation of the solution is taken into account somehow. The length scale in the solution-gradient direction, which was introduced in 2001, was intended for making sure that near solid surfaces, the element length in the surface-normal direction is selected even if that is not the minimum element length. It was also intended for making sure that in a 2D computation with a 3D mesh, there would be no dependence on the element length in the third direction. With the same objectives, and with better invariance properties, we are now introducing the direction-dependent diffusion length scale calculated from the strain-rate tensor. We accomplish those objectives, get invariance with respect to switching to a different inertial reference frame, and the element length in the surface-normal direction, even when the surface is undergoing rotation, is selected as the diffusion length scale.

Phase-field Navier–Stokes model for vesicle doublets hydrodynamics in incompressible fluid flow

In this paper, we study the hydrodynamics of a vesicle doublet suspended in an external viscous fluid flow. Vesicles in this study are modeled using the phase-field model. The bending energy and energies associated with enforcing the global volume and area are considered. In addition, the local inextensibility condition is ensured by introducing an additional equation to the system. To prevent the vesicles from overlapping, we deploy an interaction energy definition to maintain a short-range repulsion between the vesicles. The fluid flow is modeled using Navier–Stokes equations under the assumption of incompressible flows, and the vesicle evolution in time is modeled using two advection equations describing the process of advecting each vesicle by the fluid flow. Rather than solving the velocity–pressure saddle point system, we apply the Residual-Based Variational MultiScale (RBVMS) method to Navier–Stokes equations and solve the coupled systems monolithically using isogeometric analysis. We study vesicle doublet hydrodynamics in shear flow, planar extensional flow, and parabolic flow under various configurations and boundary conditions. Based on the fluid flow profile and domain configuration, we have observed various dynamics like tank-threading, locking, doublet separation, sliding, and shape transformation in tubular channels.