This paper introduces an Artificial Neural Network (ANN) augmented, Variational Multiscale (VMS) stabilized, hyper-reduced order model (HROM) for time-dependent fluid flow shape optimization applications. The method extends the authors VMS-based ROM with Discrete Empirical Interpolation Method (DEIM) hyper-reduction to achieve further computational efficiency. A fully discrete level nonlinear ANN correction term is applied to account for errors associated with both ROM projection and hyper-reduction that improves the stability, as well as the accuracy required for unsteady flows. The HROM approach we obtain here is implemented into a Sequential Least Squares Quadratic Programming (SLSQP) based gradient-based aerodynamic shape optimization problem. NACA airfoils at high angles of attack are chosen to get the transient effects like separation and vortex-shedding. The ANN correction is trained on snapshots from full-order simulations and used directly in the optimization loop at no additional cost. Numerical results show that the HROM with ANN-correction successfully captures the full order model accuracy for both transient prediction and optimization and achieves more than 95% reduction in computational cost. The results validate the effectiveness of the developed ANN-augmented HROM as an accurate and efficient counterpart for transient-CFD-based optimization problems.

We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous setting, we establish well-posedness of such concomitant formulations. Then, we propose stable and consistent finite element approximations for these underlying primal and dual problems relying on the Variational MultiScale framework. For quasi-uniform finite element partitions, we investigate approximations' general properties and establish well-posedness for two canonical choices of the sub-grid scales, i.e., the Algebraic Sub-Grid Scale and Orthogonal Sub-Grid Scale. Moreover, for continuous finite element functions, we are able to move back and forth between the discrete primal and dual formulations only by changing the design of the stabilization parameters. To conclude, we stress-test the proposed approximations through a series of progressively sophisticated cases, providing both a comparative and qualitative assessment of their numerical performance.

In this paper we consider the conforming finite element (FE) approximation of Maxwell's problem and analyse the prescription of essential boundary conditions in a weak sense using Nitsche's method. To avoid indefiniteness of the problem, the original equations are augmented with the gradient of a scalar field that allows one to impose the zero divergence of the magnetic induction, even if the exact solution for this scalar field is zero. Two FE approximations are considered, namely, one in which the approximation spaces are assumed to satisfy the appropriate inf-sup condition that render the standard Galerkin method stable, and another augmented and stabilised one that permits the use of FE interpolations of arbitrary order. Stability and convergence results are provided for the two FE formulations considered.

This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for error estimation is based on the variational multiscale (VMS) concept, where the solution is decomposed into resolved (finite element) and unresolved (sub-grid) scales. In this work, we propose an orthogonal sub-grid scale (OSGS) method for a goal-oriented error estimation in VMS discretizations. In the OSGS approach, the space of the sub-grid scales (SGSs) is orthogonal to the finite element space. The error is estimated in the quantity of interest, given by the linear functional Q(u) of the unknown u. If the SGS u' is estimated, the error in the quantity of interest can be approximated by Q(u'). Our approach is compared with a duality-based a posteriori error estimation method, which requires the solution of an additional auxiliary problem. The results indicate that both methods yield similar error estimates, whereas the VMS-based explicit approach is computationally less expensive than the duality-based implicit approach. Numerical tests demonstrated the effectiveness of our proposed error estimation techniques in terms of the quantity of interest functionals.

In this work, we describe a finite element formulation for the approximation of solid mechanics problems using a damage model under finite strain conditions. The balance equations are written in a total Lagrangian framework, employing the deviatoric component of the second Piola-Kirchhoff stress tensor, the displacement, and the pressure as primary variables. Introducing the pressure as a variable enables the treatment of incompressible materials, while incorporating the stress improves the stress approximation, which is crucial when nonlinear material laws depending on stress (or strain) are considered. In particular, we adopt the damage model proposed by Comellas et al. (International Journal for Numerical Methods in Engineering, Vol. 105, pp. 781-800, 2016), which generalizes previous isotropic damage models from infinitesimal strains to finite ones. This damage model is combined with a hyperelastic formulation for the reversible component of the deformation. The three-field formulation we consider was first introduced and analyzed for the Stokes problem by Codina (SIAM Journal on Numerical Analysis, Vol. 47, pp. 699-718, 2009). The interest of interpolating stress as an independent variable was highlighted in the work of Cervera et al. (Computer Methods in Applied Mechanics and Engineering, Vol. 199, pp. 2559-2570, 2010), and has since been successfully applied to numerous problems involving both linear and nonlinear constitutive behavior under the small strain assumption. More recently, Codina et al. (International Journal for Numerical Methods in Engineering, Vol. 125, e7540, 2024), extended the three-field formulation to geometrically nonlinear problems. The purpose of the present work is to combine these approaches, addressing problems that involve both nonlinear constitutive laws and geometrical nonlinearity with a mixed, three-field approach.

In this work, a stabilized conservative level-set method is proposed for sharp-interface modeling of two-phase flows. The level-set regularization (sharpening) equation is reformulated as a pseudo Convection-Diffusion-Reaction (CDR) problem, enabling the application of a Variational Multi-Scale (VMS)-based stabilization strategy. This approach allows the use of smaller diffusion coefficients, which greatly enhances robustness in scenarios involving large interface deformations and reduces the risk of nonphysical interface break-up. To further improve stability, the sharpening equation is solved only in a narrow band surrounding the interface, thereby avoiding one of the main instability sources in conservative level-set methods-the reliance on inadequate representation of far-field normal vectors. The employment of this localized formulation is possible within an enriched finite element framework, which naturally supports zero-thickness sharp-interface representations. Restricting the solution domain for the regularization equation not only improves numerical stability but also substantially reduces computational cost. Benchmark validations-including non-zero strain rate velocity field tests, vortex flows, Zalesak's disk problem, two and three-dimensional rising bubbles, and oscillating droplet simulations-demonstrate the framework's excellent mass conservation, accurate pressure jump representation, and reduced parasitic currents compared to the previously developed techniques in this context. Remarkably, the proposed E-FEM/CLS framework achieves these high-fidelity results even on relatively coarse meshes.

The paper presents an hp analysis of Nitsche’s method applied to Maxwell’s problem using a stabilised finite element (FE) formulation. While the FE formulation and the weak imposition of Dirichlet boundary conditions have been previously explored, the novelty of this work lies in the hp mesh refinement analysis, examining the method’s behaviour as both the mesh size h decreases and the polynomial order p increases. The study specifically aims to adapt previous analyses to account for these refinements and to highlight a slight lack of optimality in p that is intrinsic to Nitsche’s method.

In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family.

This study introduces a novel a posteriori time error indicator applied to a finite element flow solver. The proposed approach integrates stabilized finite element techniques in space and backward differentiation formula (BDF) schemes in time, adjusting time step sizes dynamically in unsteady flow simulations. The developed time error indicator and time adaptivity algorithm encompass monolithic and fractional step algorithms. In the case of fractional step algorithms, an additional term is incorporated in the error indicator to account for the error caused by the splitting. A series of numerical examples are presented to validate the reliability and robustness of the time error indicator across benchmark problems involving incompressible and isentropic compressible flows.

High-order h/p solvers in computational fluid dynamics offer scalability, efficiency, and superior error reduction compared to traditional low-order methods. Immersed boundary methods eliminate the need for body-fitted meshes but often degrade the order of the solution near boundaries, which can damage the overall accuracy of the high-order solver. This paper presents new approach to impose boundary conditions in high-order finite element or finite volume flow solvers that retain high-order P + 1 convergence, where P is the polynomial order. Furthermore, the methodology takes into account curved boundary conditions without loss in accuracy. It introduces a surrogate boundary that eliminates instabilities due to badly cut elements. We test the methodology using a high-order discontinuous Galerkin framework to solve purely elliptic problems and the compressible Navier-Stokes equations (2D and 3D), to show that we retain the formal order of convergence P + 1. Finally, we compare the results with a volume penalization approach and show that spurious pressure oscillations on the immersed boundary are eliminated when the proposed methodology is used.

This work introduces a numerical framework for addressing Fluid-Structure Interaction problems involving thin structures subject to finite strain deformations. The proposed approach utilizes an embedded mesh method to establish a coupling interface between the fluid and structural domains. The novelty of the work is the incorporation of a recently developed locking-free stabilized formulation of solid-shell elements to handle the structural domain. The framework employs established techniques to handle pressure jumps in the fluid domain across the embedding interface and enforce boundary conditions, such as discontinuous shape functions for the pressure unknowns designed to segregate nodal contributions of the cut elements, and Nitsche's method for the weak imposition of transmission conditions in the fluid. The present approach is validated through a series of benchmark cases in both 2D and 3D environments, progressively increasing in complexity. The results demonstrate good agreement with existing literature, establishing the presented framework as a viable method for addressing Fluid-Structure Interaction problems involving thin structures subject to large strains.

Los muros de mampostería son ampliamente usados en centros urbanos. El estudio de su respuesta frente a cargas explosivas es fundamental, tanto para perfeccionar el diseño de estructuras susceptibles a estas acciones, como para desarrollar herramientas de análisis de atentados, ataques militares o accidentes industriales. La predicción de la respuesta estructural de muros ante explosiones supone un desafío complejo, dada su naturaleza no lineal, influenciada por factores como la adherencia y los mecanismos de fricción entre sus componentes. En los últimos años, diversos autores han propuesto modelos de simulación con distintos niveles de complejidad. Estos incluyen micro-modelos, que detallan todos los componentes del muro, micro-modelos simplificados, donde el mortero y las interfaces se agrupan para formar superficies de contacto entre los ladrillos y macro-modelos homogéneos, que consideran un material orto trópico para reflejar las variaciones en el comportamiento mecánico en diferentes direcciones. Es esencial estudiar y comparar estos modelos, así como calibrar los numerosos parámetros que los definen. Este trabajo presenta una alternativa de modelación explícita y compara sus resultados con datos experimentales previamente obtenidos.

Wave propagation in elastodynamic problems in solids often requires fine computational meshes. In this work we propose to combine stabilized finite element methods (FEM) with an artificial neural network (ANN) correction term to solve such problems on coarse meshes. Irreducible and mixed velocity-stress formulations for the linear elasticity problem in the frequency domain are first presented and discretized using a variational multiscale FEM. A nonlinear ANN correction term is then designed to be added to the FEM algebraic matrix system and produce accurate solutions when solving elastodynamics on coarse meshes. As a case study we consider acoustic black holes (ABHs) on structural elements with high aspect ratios such as beams and plates. ABHs are traps for flexural waves based on reducing the structural thickness according to a power-law profile at the end of a beam, or within a two-dimensional circular indentation in a plate. For the ABH to function properly, the thickness at the termination/center must be very small, which demands very fine computational meshes. The proposed strategy combining the stabilized FEM with the ANN correction allows us to accurately simulate the response of ABHs on coarse meshes for values of the ABH order and residual thickness outside the training test, as well as for different excitation frequencies.

This paper presents a dynamic formulation for the simulation of nearly incompressible structures using a mixed finite element method with equal-order interpolation pairs. Specifically, the nodal unknowns are the displacement and the volumetric strain component, something that makes possible the reconstruction of the complete stain at the integration point level and thus enables the use of strain-driven constitutive laws. Furthermore, we also discuss the resulting eigenvalue problem and how it can be applied for the modal analysis of linear elastic solids. The article puts special emphasis on the stabilisation technique used, which becomes crucial in the resolution of the generalised eigenvalue problem. In particular, we prove that using a variational multiscale method assuming the sub-grid scales to lie in the finite element space orthogonal to that of the approximation, namely the Orthogonal Sub-Grid Scales (OSGS), results in a convenient linear and symmetric generalised eigenvalue problem. The correctness, convergence and performance of the method are proven by solving a series of two- and three-dimensional examples.

In this work, an algorithm for topology optimization of incompressible structures is proposed, in both small and finite strain assumptions and in which the loads come from the interaction with a surrounding fluid. The algorithm considers a classical block-iterative scheme, in which the solid and the fluid mechanics problems are solved sequentially to simulate the interaction between them. Several stabilized mixed finite element formulations based on the Variational Multi-Scale approach are considered to be capable of tackling the incompressible limit for the numerical approximation of the solid. The fluid is considered as an incompressible Newtonian fluid flow which is combined with an Arbitrary-Lagrangian Eulerian formulation to account for the moving part of the domain. Several numerical examples are presented and discussed to assess the robustness of the proposed algorithm and its applicability to the topology optimization of incompressible elastic solids subjected to Newtonian incompressible fluid loads.

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.

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.

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.

This paper presents mixed finite element formulations to approximate the hyperelasticity problem using as unknowns the displacements and either stresses or pressure or both. These mixed formulations require either finite element spaces for the unknowns that satisfy the proper inf‐sup conditions to guarantee stability or to employ stabilized finite element formulations that provide freedom for the choice of the interpolating spaces. The latter approach is followed in this work, using the Variational Multiscale concept to derive these formulations. Regarding the tackling of the geometry, we consider both infinitesimal and finite strain problems, considering for the latter both an updated Lagrangian and a total Lagrangian description of the governing equations. The combination of the different geometrical descriptions and the mixed formulations employed provides a good number of alternatives that are all reviewed in this paper.

We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell-Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method we consider consists of an augmented formulation with the introduction of a mesh dependent term that can be regarded as the Laplacian of the multiplier of the divergence constraint. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems. We investigate the numerical performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.