The
statistical
operators typically applied in postprocessing numerical databases for statistically steady turbulence are a mixture of physical averages in homogeneous spatial directions and in time. ...Alternative averaging operators may involve phase or ensemble averages over different simulations of the same flow. In this paper, we propose straightforward metrics to assess the relative importance of these averages, employing a mixed averaging analysis of the variance. We apply our novel indicators to two statistically steady turbulent flows that are homogeneous in the spanwise direction. In addition, this study highlights the local effectiveness of the averaging operator, which can vary significantly depending on the mean flow velocity and turbulent length scales. The work can be utilized to identify the most effective averaging procedure in flow configurations featuring at least two homogeneous directions. Thus, this will contribute to achieving better statistics for turbulent flow predictions or reducing computing time.
Summary
We present a computational framework for the simulation of J2‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid ...spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.
We propose a new unfitted/immersed computational framework for nonlinear solid mechanics, which bypasses the complexities associated with the generation of CAD representations and subsequent ...body-fitted meshing. This approach allows to speed up the cycle of design and analysis in complex geometry and requires relatively simple computer graphics representations of the surface geometries to be simulated, such as the Standard Tessellation Language (STL format). Complex data structures and integration on cut elements are avoided by means of an approximate boundary representation and a modification (shifting) of the boundary conditions to maintain optimal accuracy. An extensive set of computational experiments in two and three dimensions is included.
Here, we propose a new unfitted/immersed computational framework for nonlinear solid mechanics, which bypasses the complexities associated with the generation of CAD representations and subsequent ...body-fitted meshing. This approach allows to speed up the cycle of design and analysis in complex geometry and requires relatively simple computer graphics representations of the surface geometries to be simulated, such as the Standard Tessellation Language (STL format). Complex data structures and integration on cut elements are avoided by means of an approximate boundary representation and a modification (shifting) of the boundary conditions to maintain optimal accuracy. An extensive set of computational experiments in two and three dimensions is included.
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 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 L2-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 complement these theoretical developments with numerical experiments in two and three dimensions.
•A novel approximate domain (unfitted) method for the Poisson and Stokes operators.•The Shifted Boundary Method (SBM) does not require cut cells.•The SBM is defined on a surrogate domain, with modified boundary conditions.•A simpler SBM, without restrictive assumptions and unnecessary stabilization terms.•Improved proofs of stability, convergence and L2-error estimates.
•A new algorithm for multiplicative nonlinear deviatoric viscoelasticity.•Stable computations on piecewise linear finite elements on triangular and tetrahedral grids.•Nearly and fully incompressible ...materials.•Variational multiscale stabilization scaled with the viscous energy dissipation.
We present a computational approach to solve problems in multiplicative nonlinear viscoelasticity using piecewise linear finite elements on triangular and tetrahedral grids, which are very versatile for simulations in complex geometry. Our strategy is based on (1) formulating the equations of mechanics as a mixed first-order system, in which a rate form of the pressure equation is utilized in place of the standard constitutive relationship, and (2) utilizing the variational multiscale approach, in which the stabilization parameter is scaled with the viscous energy dissipation.
We introduce and analyze a penalty-free formulation of the Shifted Boundary Method (SBM), inspired by the asymmetric version of the Nitsche method. We prove its stability and convergence for ...arbitrary order finite element interpolation spaces and we test its performance with a number of numerical experiments. Moreover, while the SBM was previously believed to be only asymptotically consistent (in the sense of Galerkin orthogonality), we prove here that it is indeed exactly consistent.
The shifted boundary method for solid mechanics Atallah, Nabil M.; Canuto, Claudio; Scovazzi, Guglielmo
International journal for numerical methods in engineering,
30 October 2021, 2021-10-30, Volume:
122, Issue:
20
Journal Article
Peer reviewed
We propose a new embedded/immersed framework for computational solid mechanics, aimed at vastly speeding up the cycle of design and analysis in complex geometry. In many problems of interest, our ...approach bypasses the complexities associated with the generation of CAD representations and subsequent body‐fitted meshing, since it only requires relatively simple representations of the surface geometries to be simulated, such as collections of disconnected triangles in three dimensions, widely used in computer graphics. Our approach avoids the complex treatment of cut elements, by resorting to an approximate boundary representation and a special (shifted) treatment of the boundary conditions to maintain optimal accuracy. Natural applications of the proposed approach are problems in biomechanics and geomechanics, in which the geometry to be simulated is obtained from imaging techniques. Similarly, our computational framework can easily treat geometries that are the result of topology optimization methods and are realized with additive manufacturing technologies. We present a full analysis of stability and convergence of the method, and we complement it with an extensive set of computational experiments in two and three dimensions, for progressively more complex geometries.
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