Variational Multiscale Framework Research Articles

Stress–displacement stabilized finite element analysis of thin structures using Solid-Shell elements, Part II: Finite strain hyperelasticity

This work is the second of a two-part research project focused on modeling solid-shell elements using a stabilized two-field finite element formulation. The first part introduces a stabilization technique based on the Variational Multiscale framework, which is proven to effectively address numerical locking in infinitesimal strain problems. The primary objective of the study was to characterize the inherent numerical locking effects of solid-shell elements in order to comprehensively understand their triggers and how stabilized mixed formulations can overcome them. In this current phase of the work, the concept is extended to finite strain solid dynamics involving hyperelastic materials. The aim of introducing this method is to obtain a robust stabilized mixed formulation that enhances the accuracy of the stress field. This improved formulation holds great potential for accurately approximating shell structures undergoing finite deformations. To this end, three techniques based in the Variational Multiscale stabilization framework are presented. These stabilized formulations allow circumventing the compatibility restriction of interpolating spaces of the unknowns inherent to mixed formulations, thus allowing any combination of them. The accuracy of the stress field is successfully enhanced while maintaining the accuracy of the displacement field. These improvements are also inherited to the solid-shell elements, providing locking-free approximation of thin structures.

A Stabilized Finite Element Framework for Anisotropic Adaptive Topology Optimization of Incompressible Fluid Flows

This paper assesses the feasibility of performing topology optimization of laminar incompressible flows governed by the steady-state Navier–Stokes equations using anisotropic mesh adaptation to achieve a high-fidelity description of all fluid–solid interfaces. The present implementation combines an immersed volume method solving stabilized finite element formulations cast in the variational multiscale (VMS) framework and level-set representations of the fluid–solid interfaces, which are used as an a posteriori anisotropic error estimator to minimize interpolation errors under the constraint of a prescribed number of nodes in the mesh. Numerical results obtained for several two-dimensional problems of power dissipation minimization show that the optimal designs are mesh-independent (although the convergence rate does decreases as the number of nodes increases), agree well with reference results from the literature, and provide superior accuracy over prior studies solved on isotropic meshes (fixed or adaptively refined).

Large-scale parallel topology optimization of three-dimensional incompressible fluid flows in a level set, anisotropic mesh adaptation framework

This papers considers the topology optimization of duct flows governed by the three-dimensional steady state Navier–Stokes equations, using anisotropic mesh adaptation to achieve a high-fidelity description of all fluid–solid interfaces. The numerical framework combines an immersed volume method solving stabilized, linear equal-order finite element formulations cast in the Variational Multiscale (VMS) framework, and level set representations of the interface, used as a posteriori anisotropic error estimator to minimize the interpolation error under the constraint of a prescribed number of nodes in the mesh. Both the resolution and remeshing steps are performed in a massively parallel framework allowing for the optimization of large-scale systems. In particular, an original parallelization strategy is used for mesh adaptation, that combines local remeshing performed sequentially and independently on each subdomain with blocked interfaces, and constrained repartitioning to optimally move the interfaces between subdomains in an optimal way, both iterated until a satisfying mesh and partition are obtained. The proposed approach reduces the computational burden related to the call of the finite element solver, compared to classical optimization schemes working on uniform grids with similar mesh refinement. For a given number of nodes, it improves the accuracy in the geometric description of all layouts. Finally, it has the potential to alleviates the end user from most of the post-processing step aiming at extracting the final layout, due to ability of anisotropic adapted meshes to generate intrinsically smooth designs. Numerical results are provided for several three-dimensional problems of power dissipation minimization involving several dozen million state degrees of freedom, for which the optimal designs agree well with reference results from the literature, while providing superior accuracy over prior studies solved on isotropic meshes (in the sense that the flow is better resolved, especially in the near-wall regions, and the layouts are more smooth). The potential of the method for engineering problems of practical interest is eventually exposed by optimizing the distributor section conveying the cold fluid within the plates of a plate fin heat exchanger.

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).

Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling

We propose a new physics guided machine learning (PGML) paradigm that leverages the variational multiscale (VMS) framework and available data to dramatically increase the accuracy of reduced order models (ROMs) at a modest computational cost. The hierarchical structure of the ROM basis and the VMS framework enable a natural separation of the resolved and unresolved ROM spatial scales. Modern PGML algorithms are used to construct novel models for the interaction among the resolved and unresolved ROM scales. Specifically, the new framework builds ROM operators that are closest to the true interaction terms in the VMS framework. Finally, machine learning is used to reduce the projection error and further increase the ROM accuracy. Our numerical experiments for a two-dimensional vorticity transport problem show that the novel PGML-VMS-ROM paradigm maintains the low computational cost of current ROMs while significantly increasing the ROM accuracy.

Stabilized finite element methods for a fully-implicit logarithmic reformulation of the Oldroyd-B constitutive law

Logarithmic conformation reformulations for viscoelastic constitutive laws have alleviated the high Weissenberg number problem, and the exploration of highly elastic flows became possible. However, stabilized formulations for logarithmic conformation reformulations in the context of finite element methods have not yet been widely studied. We present stabilized formulations for the logarithmic reformulation of the Oldroyd-B model by Saramito (2014) based on the Variational Multiscale framework and Galerkin/Least-Squares method. The reformulation allows the use of Newton’s method due to its fully-implicit nature for solving the steady-state problem directly. The proposed stabilization methods cure instabilities of convection and compatibility while preserving a three-field problem. The spatial accuracy of the formulations is assessed with the four-roll periodic box, revealing comparable accuracy between the methods. The formulations are validated with benchmark flows past a cylinder and through a 4:1 contraction. We found an excellent agreement to benchmark results in the literature. The algebraic sub-grid scale method is highly robust, indicated by the high limiting Weissenberg numbers in the benchmark flows.

Variational coupling of non-matching discretizations across finitely deforming fluid-structure interfaces.

This paper presents a stabilized monolithic method for coupling incompressible viscous fluids with finitely deforming elastic solids across non-matching interfacial meshes. Governing equations for the solid are written in the finite deformation Lagrangian frame using velocity field as the primary unknown, while the governing equations for the fluid are written in an Arbitrary Lagrangian-Eulerian (ALE) frame to accommodate large motions of the fluid-solid interfaces. Interface coupling terms are derived by embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework and locally resolving the fine-scale variational equations from the fluid and the solid subdomains along the common interface. The unique attribute of the proposed Variational Multiscale Discontinuous Galerkin (VMDG) method is a systematic procedure for deriving analytical expression for the traction Lagrange multiplier at the fluid-solid interface. The structure of the interface stabilization tensor emerges naturally and is shown to be a function of the boundary operators that are associated with the domain interior operators from the fluid and the solid subdomains. The derived stabilization tensor possesses the features of area-averaging and stress-averaging and evolves spatially and temporally with the evolving nonlinear fields at the interface. An analysis of the mathematical properties of the interface stabilization tensor is presented and subsequently verified numerically. The method is implemented using four-node tetrahedral elements for the fluid as well as the solid subdomains while employing equal-order interpolations for the various interacting fields. Benchmark problems are presented for the verification of the method for finitely deforming fluid-structure interfaces.

A VMS–based fractional step technique for the compressible Navier–Stokes equations using conservative variables

In this paper we address the compressible Navier–Stokes equations written in the so-called conservative formulation. In particular, we focus on the possibility of uncoupling the computation of the problem unknowns, namely, density, linear momentum and total energy, a technique usually labeled as fractional step method, which allows to reduce the associated computational cost. The proposed methodology is a finite-element solver supplemented with a stabilization technique within the Variational Multi-Scale framework. In this regard, we consider orthogonal and dynamic definitions for the subscales. This discretization in space shows an adequate stability, permitting in particular the use of equal interpolation for all variables in play. However, we complement it with a shock-capturing operator in order to solve problems involving shocks. Several representative benchmark flow simulations are performed, which demonstrate the suitability of the proposed algorithm for a vast range of regimes.

A variational multiscale immersed meshfree method for fluid structure interactive systems involving shock waves

We develop an immersed meshfree method under a variational multiscale framework for modeling fluid–structure interactive systems involving shock waves. The proposed method enables flexible non-body-fitted discretization, approximations, and quadrature rules for solid and fluid subdomains. The interfacial compatibility conditions are imposed by a volumetric constraint, which avoids the tedious contour integral and interface tracking. The reproducing kernel particle method (RKPM) is employed for both solid and fluid sub-systems, which allows arbitrary control of the orders of continuity and approximation, as well as flexibility in discretization, making it particularly advantageous for modeling fluid–structure interaction (FSI). In the proposed approach, the fictitious fluid is combined with the foreground solid, forming an “effective solid problem” solved on a moving foreground domain, while the background fluid problem is solved with prescribed solid velocity in the overlapping domain to reduce the leaking instability and mesh sensitivity. The variational multiscale immersed method (VMIM) is employed to enhance accuracy and stability in FS coupling, which leads to a residual-based stabilization. The MUSCL-SCNI shock algorithm provides a natural way of introducing the Riemann solution in the shock algorithm via the SCNI contour integral for desirable accuracy. The employment of SCNI in the proposed framework also provides computational efficiency, accuracy, and stability. Using a larger RKPM support size in the fluid domain can effectively suppress the leaking instability. The effectiveness of the proposed methods is verified in solving several FSI problems with shock waves, and the enhanced stability and accuracy of the proposed methods compared to the classical immersed approach have also been demonstrated.

A higher-order finite element method with unstructured anisotropic mesh adaption for two phase flows with surface tension

A novel finite element framework is proposed for the numerical simulation of two phase flows with surface tension. The Level-Set (LS) method with piece-wise quadratic (P2) interpolation for the liquid–gas interface is used in order to reach higher-order convergence rates in regions with smooth interface. A balanced-force implementation of the continuum surface force model is used to take into account the surface tension and to solve static problems as accurately as possible. Given that this requires a balance between the discretization used for the LS function, and that used for the pressure field, an equal-order P2/P2/P2 scheme is proposed for the Navier–Stokes and LS advection equations, which are strongly coupled with each other. This fully implicit formulation is stabilized using the residual-based variational multiscale framework. In order to improve the accuracy and obtain optimal convergence rates with a minimum number of elements, an anisotropic mesh adaption method is proposed where the unstructured mesh is kept as fine as possible close to the zero iso-value of the P2 LS function. Elements are automatically stretched in regions with flat interface in order to keep the complexity fixed during the simulation. The accuracy and efficiency of this approach are demonstrated for two and three dimensional simulations of a rising bubble.

Nitsche’s method as a variational multiscale formulation and a resulting boundary layer fine-scale model

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 fine scales at the Dirichlet boundaries. This results in an additional boundary term with a new model parameter. We then propose a parameter estimation strategy for all parameters involved that is also consistent for higher-order basis functions. We illustrate with numerical experiments that our new augmented model mitigates the overly diffusive behavior that the classical residual-based fine-scale model exhibits in boundary layers at boundaries with weakly enforced essential conditions.

Variationally derived closure models for large eddy simulation of incompressible turbulent flows

AbstractWe present a variationally consistent method for deriving residual‐based closure models for incompressible Navier–Stokes equations. The method is based on the fine‐scale variational structure facilitated by the variational multiscale framework where fine scales are driven by the residuals of the Euler–Lagrange equations of the resolved scales in the balance of momentum and conservation of mass equations. A bubble‐functions based approach is applied directly to the fine‐scale variational equation to derive analytical expressions for the closure model. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the iterative solution strategy for the nonlinear equations. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain. Numerical tests with the proposed model are presented for various classes of turbulent flow problems to show its generality and range of applicability. The test cases investigated include Taylor–Green vortex stretching, statistically stationary wall‐bounded channel flows, and modeling the effects of the geometry of the leading edge of the plate on the instability of the boundary layer that leads to flow separation and flow reversal over flat plates of finite thickness.

Development of an algebraic fractional step scheme for the primitive formulation of the compressible Navier-Stokes equations

In this work we address the compressible Navier-Stokes equations written in the so-called primitive formulation. The proposed methodology is a finite-element solver based on a fractional step scheme in time, which allows to uncouple the calculation of the problem unknowns providing important savings in computational cost. In addition, we include a stabilization technique within the Variational Multi-Scale framework and, in particular, we consider orthogonal and dynamic definitions for the subscales. In order to overcome any wave reflections which may arise in aeroacoustic simulations at the low compressibility regime, we present a method for enforcing boundary conditions based on a combination of a zero order non-reflecting condition plus the weak imposition of Dirichlet boundary conditions over the external contours. Several representative benchmark flow simulations are performed, which demonstrate the suitability of the proposed algorithm for the subsonic regime.

Weakly imposed boundary conditions for shear-rate dependent non-Newtonian fluids: application to cardiovascular flows.

This paper presents a stabilized formulation for the generalized Navier-Stokes equations for weak enforcement of essential boundary conditions. The non-Newtonian behavior of blood is modeled via shear-rate dependent constitutive equations. The boundary terms for weak enforcement of Dirichlet boundary conditions are derived via locally resolving the fine-scale variational equation facilitated by the Variational Multiscale (VMS) framework. The proposed method reproduces the consistency and stabilization terms that are present in the Nitsche type approaches. In addition, for the shear-rate fluids, two more boundary terms appear. One of these terms is the viscosity-derivative term and is a function of the shear-rate, while the other term is a zeroth-order term. These terms play an important role in attaining optimal convergence rates for the velocity and pressure fields in the norms considered. A most significant contribution is the form of the stabilization tensors that are also variationally derived. Employing edge functions the edge stabilization tensor is numerically evaluated, and it adaptively adjusts itself to the magnitude of the boundary residual. The resulting formulation is variationally consistent and the weakly imposed no-slip boundary condition leads to higher accuracy of the spatial gradients for coarse boundary-layer meshes when compared with the traditional strongly imposed boundary conditions. This feature of the present approach will be of significance in imposing interfacial continuity conditions across non-matching discretizations in blood-artery interaction problems. A set of test cases is presented to investigate the mathematical attributes of the method and a patient-specific case is presented to show its clinical relevance.

Variational multiscale method for fully coupled thermomechanical interface contact and debonding problems

In this study, a computational framework is proposed for thermomechanical contact and debonding problems with proper thermal resistance at the interface. Using the Variational Multiscale (VMS) framework, we present a fully coupled thermomechanical formulation with an explicit expression of the pressure at the contact interface. The formulation considers the quasi-static balance of the momentum and the transient heat transfer problem in a fully coupled fashion. At the interface, two different contact constitutive models are utilized for tension and compression. For tensile problems, in the mechanical phase, a tensile debonding model is employed, whereas in the thermal phase, the displacement-dependent model is employed. For compressive problems, in the mechanical phase, a Coulomb frictional model is employed while in the thermal phase, a pressure-dependent model is embedded. Because of the naturally derived interface stability terms that possess area- and stress-weighting, the proposed VMS formulation accommodates contact/debonding and contact/frictional sliding at the interface due to both thermal and mechanical loading without losing numerical stability. The proposed method is applied to a class of numerical test problems with discontinuity at the interfaces, and good agreement with analytical and numerical data is achieved.

Modeling of steep layers in singularly perturbed diffusion–reaction equation via flexible fine-scale basis

We present a new stabilized method for the diffusion–reaction equation which develops sharp boundary and/or internal layers for the reaction-dominated case (i.e. singularly perturbed case). The method relies on an improved expression for the stabilization parameter that is derived via the fine-scale variational formulation facilitated by the variational multiscale (VMS) framework. The proposed fine-scale basis consists of enrichment functions which may be nonzero at element edges. The derived stabilization parameter enjoys spatial variation over element interiors and along inter-element boundaries that helps model the rapid variation of the residual of the Euler–Lagrange equations over the domain. This feature also facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects. The method is able to better satisfy the maximum principle as compared to other existing methods. We present a priori mathematical analysis of the stability and convergence of the method for the diffusion–reaction equation. Optimal convergence on meshes comprised of linear triangles and bilinear quadrilateral elements are presented for smooth problems, as well as for problems with steep boundary layers. Stability and accuracy features of the method for problems with discontinuous forcing function, internal layers, and boundary layers are shown and its performance on unstructured and distorted meshes comprised of quadrilateral and triangular elements is highlighted.

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.

Element boundary terms in reduced order models for flow problems: Domain decomposition and adaptive coarse mesh hyper-reduction

In this paper we present a finite-element based reduced order model and, in particular, we consider two aspects related to the introduction of inter-element boundary terms in the formulation. The first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non-matching meshes and discontinuous physical properties. The second is a coarse mesh hyper-reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms. As the finite element full order model, the reduced order model is based on the Variational Multi-Scale framework, with sub-grid scales defined not only in the element interiors, but also on the inter-element boundaries. We present some examples of application using the incompressible Navier–Stokes equations and the Boussinesq approximation.

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.

A residual-based variational multiscale method with weak imposition of boundary conditions for buoyancy-driven flows

There is growing interest in high fidelity simulations of buoyancy-driven flows in indoor environments. This is driven to a large extent by the push to utilize naturally occurring flows to reduce energy consumption in the built environment. Naturally occurring (buoyancy-driven) flows exhibit spatiotemporal thermal fluctuations, and accurately capturing these thermal variations is essential to calculate energy usage. In this work, we deploy a residual-based variational multiscale (VMS) finite element large-eddy simulation model for accurately modeling buoyancy-driven flows in enclosed environments. We use the canonical example of the Rayleigh–Bénard convection problem in 2D and 3D to verify and validate the VMS model. We show good comparison with benchmark numerical and experimental results across seven orders of magnitude variation in Rayleigh numbers (Ra∼103 to 1010), covering laminar, transition, and turbulent regimes without any extra treatments. We additionally employ weak enforcement of Dirichlet boundary conditions for both velocity and temperature, and show that comparable results can be produced with much coarser meshes. This confirms that the VMS framework with weak imposition of Dirichlet boundary conditions is a computationally efficient approach to model buoyancy-driven flow physics in complex indoor environments.