The Second-Generation Shifted Boundary Method and Its Numerical Analysis

Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that {\it shifts} the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced $L^{2}$-estimates without the cumbersome assumption - of earlier proofs - that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We present numerical experiments in two and three dimensions.Comment: 28 pages, 6 figures, 4 table

A study on the statistical convergence of turbulence simulations around a cylinder

The turbulent flow around a circular cylinder at a Reynolds number equal to 3900 is studied by an implicit Large Eddy Simulation performed by means of a discontinuous Galerkin finite element solver. The average velocity field in the wake is evaluated and compared with experimental data from the literature. The focus of the present work is on the estimation of the statistical uncertainty which is related to the use of a finite time window for the averaging operation. This topic represents an open problem for both Direct Numerical Simulations and Large Eddy Simulations in which it is difficult to define a priori the size of the time window which gives statistically converged averaged quantities. Different techniques to estimate this uncertainty are compared in order to get a quantitative criterion for checking the convergence of statistics. In particular, the Non-Overlapping Batch Means and the Batch Means Batch Correlations techniques are applied to the present test case

Weak Boundary Conditions for Lagrangian Shock Hydrodynamics: A High-Order Finite Element Implementation on Curved Boundaries

We propose a new Nitsche-type approach for weak enforcement of normal velocity boundary conditions for a Lagrangian discretization of the compressible shock-hydrodynamics equations using high-order finite elements on curved boundaries. Specifically, the variational formulation is appropriately modified to enforce free-slip wall boundary conditions, without perturbing the structure of the function spaces used to represent the solution, with a considerable simplification with respect to traditional approaches. Total energy is conserved and the resulting mass matrices are constant in time. The robustness and accuracy of the proposed method are validated with an extensive set of tests involving nontrivial curved boundaries

Vers des méthodes immerées generalisées:une approche Shifted Boundary P1 avec des flux d'ordre pour les équations de Darcy

In this paper, we propose to extend the recent embedded boundary method known as "shifted boundary method" to the Darcy flow problems. The aim is to provide an improved formulation that would give, using linear approximation, at least second order accuracy on bothflux and pressure variables, for any kind of boundary condition, considering embedded simulations.The strategy adopted here is to enrich the approximation of the pressure using Taylor expansionsalong the edges. The objective of this enrichment is to give a quadratic shape to the pressure. The resulted scheme provides high order accuracy on both variables for embedded simulations with an overall second order accuracy, that is bumped to third order for the pressure when only Dirichletboundaries are embedded

The shifted fracture method

This is the peer reviewed version of the following article: Li, K. [et al.]. The shifted fracture method. "International journal for numerical methods in engineering", 30 Novembre 2021, vol. 122, núm. 22, p. 6641-6679, which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/nme.6806. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.We propose a new framework for fracture mechanics, based on the idea of an approximate fracture geometry representation combined with approximate interface conditions. Our approach evolves from the shifted interface method, and introduces the concept of an approximate fracture surface composed of the full edges/faces of an underlying grid that are geometrically close to the true fracture geometry. The original interface conditions are then modified on the surrogate fracture geometry, by way of Taylor expansions. The shifted fracture method does not require cut cell computations or complex data structures, since the behavior of the true fracture is mimicked with standard integrals on the approximate fracture surface. Furthermore, the energetics of the true fracture are represented within the accuracy of the underlying polynomial finite element approximation and independently of the grid topology. The computational framework is presented here in its generality and then applied in the specific context of cohesive zone models, with an extensive set of numerical experiments in two and three dimensions.Peer ReviewedPostprint (author's final draft

Shifted boundary method pour systèmes hyperboliques: ondes linéaires et équations shallow water

We propose a new computational approach for embedded boundary simulations ofhyperbolic systems. Applications are shown for the linear wave equations and for the nonlinearshallow water system. The proposed approach belongs to the class of surrogate/approximateboundary algorithms and is based on the idea of shifting the location where boundary conditionsare applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforcedweakly, are appropriately modified to preserve optimal error convergence rates. This frameworkis applied here in the setting of a stabilized finite element method, even though other spatialdiscretization techniques could have been employed. Accuracy, stability and robustness of theproposed method are tested by means of an extensive set of computational experiments for theacoustic wave propagation equations and shallow water equations. Comparisons with standardweak boundary conditions imposed on grids that conform to the geometry of the computationaldomain boundaries are also presented.On propose une nouvelle approche pour des simulations avec bords immergés pour des systèmes hyperboliques et en particulier les équations shallow water. L’approche proposée consiste en modifier les conditions au bords avec un développement limité permettant d’assurer l’ordre deux avec des embedded boundaries. L’approche est implementé est ici dans le cadre d’une méthode de type stabilized finite element sur un très grand nombre de cas tests représentatifs d’applications de propagation de vagues et inondatio

A Weighted Shifted Boundary Method for Free Surface Flows

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods and was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations, and many additional problems [1, 2]. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. We extend here the SBM to the simulation of incompressible Navier-Stokes flows with moving free-surfaces, by appropriately weighting its variational form with the elemental volume fraction of active fluid [3]. This approach prevents spurious pressure oscillations in time, which would otherwise be produced if the total active fluid volume were to change abruptly over a time step. In fact, the proposed weighted SBM method induces small mass (i.e., volume) conservation errors, which converge quadratically in the case of piecewise-linear finite element interpolations, as the grid is refined. We present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method. References [1] Alex Main and Guglielmo Scovazzi. The Shifted Boundary Method for embedded domain com- putations. Part I: Poisson and Stokes problems. Journal of Computational Physics, 372:972–995, 2018. [2] Alex Main and Guglielmo Scovazzi. The Shifted Boundary Method for embedded domain computations. Part II: Linear advection–diffusion and incompressible Navier–Stokes equations. Journal of Computational Physics, 372:996–1026, 2018. [3] O Colomés, AG Main, L Nouveau, G Scovazzi. A Weighted Shifted Boundary Method for Free Surface Flow Problems. Journal of Computational Physics, doi: https://doi.org/10.1016/j.jcp.2020.109837. The support of the U.S. Department of Energy, Office of Science, the U.S. Office of Naval Research, and ExxonMobil Upstream Research Company (Houston, TX) are gratefully acknowledged

A multi-scale Q1/P0 approach to langrangian shock hydrodynamics.

A new multi-scale, stabilized method for Q1/P0 finite element computations of Lagrangian shock hydrodynamics is presented. Instabilities (of hourglass type) are controlled by a stabilizing operator derived using the variational multi-scale analysis paradigm. The resulting stabilizing term takes the form of a pressure correction. With respect to currently implemented hourglass control approaches, the novelty of the method resides in its residual-based character. The stabilizing residual has a definite physical meaning, since it embeds a discrete form of the Clausius-Duhem inequality. Effectively, the proposed stabilization samples and acts to counter the production of entropy due to numerical instabilities. The proposed technique is applicable to materials with no shear strength, for which there exists a caloric equation of state. The stabilization operator is incorporated into a mid-point, predictor/multi-corrector time integration algorithm, which conserves mass, momentum and total energy. Encouraging numerical results in the context of compressible gas dynamics confirm the potential of the method