SEARCH · Engineering Papers
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
This is the final report on the project, and describes the main accomplishments in the development of the Shifted Boundary Method (SBM), a new computational framework for embedded/immersed computational mechanics applications.
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 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 complement these theoretical developments with numerical experiments in two and three dimensions.
The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.
Vortex interactions and their role in turbulent flow are examined. The objectives are twofold. First, to use the existing axisymmetric code to study the annihilation process of colliding vortex rings and determine the relevance of this problem to similar 3-D phenomena. The second objective is to extend the code to three dimensions. The code under development is unique in that it can compute flows in a truly infinite domain (i.e., without periodic boundary conditions or approximations from truncating the domain). Because of this, the far field sound can be computed, and therefore, contribute to improved models of turbulence generated noise for this class of flows. Issues which can be addressed by the code include: effects of viscosity on mode selection in azimuthal breakdown of vortex rings (i.e., the Widnall instability); reconnection, the associated production of small scales, and the time scale of the process.
The asymptotic solutions of Navier-Stokes equations for vortex filaments of finite strength with small effective vortical cores are summarized with special emphasis placed on the physical meaning and the practical limit to the applicability of the asymptotic solution. Finite-difference solutions of Navier-Stokes equations for the marging of the filament(s) are described with a focus on the development of the approximate boundary conditions for the computational domain. An efficiency study employing a model problem is used to assess the advantages of the present approximate boundary condition method over previously used techniques. Applications of the present method are presented for the motion and decay of a 3:1 elliptic vortex ring, and for the merging process of a pair of coaxial vortex rings. A numerical procedure for the problem of local merging of vortex filaments, which requires the asymptotic analysis as well as the numerical Navier-Stokes solver, is also presented.
The identification of the geometrical structure of the system boundary for a two-dimensional diffusion system is considered. The domain identification treated is converted into an optimization problem based on a fit-to-data criterion and theoretical convergence results for approximate identification techniques are discussed. Results of numerical experiments to demonstrate the efficiency of the theoretical ideas are reported.
Finite-difference Navier-Stokes calculations for unsteady, three-dimensional, incompressible, viscous flows induced by initial vorticity distributions are presented and discussed in this paper. The initial vorticity distributions are assumed to be embedded in a flow field of infinite extent that is quiescent at infinity. These vorticity distributions are typical of vortex rings and other closed vortical tubes or structures. Such structures are important elements in fluid flows such as jets, atmospheric convection and the far-field wakes of aircraft; studies of their interaction may aid in an understanding of complex fluid flows. The calculations employ a method recently proposed by Ting to approximate the infinite-domain boundary value problem with a finite boundary computational domain, and this method is shown to yield accurate three-dimensional results for reasonable expenditures of computer time. Because of the efficiency of the boundary condition technique and the resulting Navier-Stokes code, a 16-bit minicomputer with virtual memory was capable of performing the calculations for the unsteady motion of two obliquely colliding vortex rings. The results of these calculations are presented in the paper.
We present results from three-dimensional magnetohydrodynamic simulations of the emergence of a twisted convection zone flux tube into a pre-existing coronal dipole field. As in previous simulations, following the partial emergence of the sub-surface flux into the corona, a combination of vortical motions and internal magnetic reconnection forms a coronal flux rope. Then, in the simulations presented here, external reconnection between the emerging field and the pre-existing dipole coronal field allows further expansion of the coronal flux rope into the corona. After sufficient expansion, internal reconnection occurs beneath the coronal flux rope axis, and the flux rope erupts up to the top boundary of the simulation domain (approximately 36 Mm above the surface).We find that the presence of a pre-existing field, orientated in a direction to facilitate reconnection with the emerging field, is vital to the fast rise of the coronal flux rope. The simulations shown in this paper are able to self-consistently create many of the surface and coronal signatures used by coronal mass ejection (CME) models. These signatures include surface shearing and rotational motions, quadrupolar geometry above the surface, central sheared arcades reconnecting with oppositely orientated overlying dipole fields, the formation of coronal flux ropes underlying potential coronal field, and internal reconnection which resembles the classical flare reconnection scenario. This suggests that proposed mechanisms for the initiation of a CME, such as "magnetic breakout," are operating during the emergence of new active regions.
The asymptotic solutions of Navier-Stokes equations for vortex filaments of finite strength with small effective vortical cores are summarized. Emphases are placed on the physical meaning and the practical limit to the applicability of the asymptotic solution. Finite-difference solutions of Navier-Stokes equations for the merging of the filament(s) are described. It is focused on the development of the approximate boundary conditions for the computational domain.
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. 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 extend 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. 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. Finally, we present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method.
The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. The surrogate domain is constructed so as to avoid cut cells and the associated problematic implementation and numerical integration issues. 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. Here, in this article, we extend the SBM to the simulation of incompressible Stokes flow, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach allows to drastically reduce spurious pressure oscillations in time, which are produced if the total volume of active fluid were to change abruptly over a time step. The proposed Weighted SBM (W-SBM) exactly preserves states of hydrostatic equilibrium, and induces small mass and momentum conservation errors, which converge as the grid is refined. This is in analogy to cutFEMs and related unfitted approaches, which rely on an affine representation of cut boundaries. We demonstrate the robustness and accuracy of the proposed method with an extensive suite of two-dimensional tests.
We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.
The structure of transient scales formed on pure and Y or Zr-doped Ni-15Cr-13Al alloys oxidized for 0.1 hr at 1100 C was studied by the use of transmission electron microscopy. Crystallographically oriented scales were found on all three alloys, but especially for the Zr-doped NiCrAl. The oriented scales consisted of alpha-(Al,Cr)2O3, Ni(Al,Cr)2O4 and gamma-Al2O3. They were often found in intimate contact with each other such that the close-packed planes and directions of one oxide phase were aligned with those of another. The prominent structural features of the oriented scales were approximately equal to micrometer subgrains; voids, antiphase domain boundaries and aligned precipitates were also prevalent. Randomly oriented alpha-Al2O3 was also found and was the only oxide ever observed at the immediate oxide metal interface. These approximately 0.15 micrometer grains were populated by intragranular voids which decreased in size and number towards the oxide metal interface. A sequence of oxidation was proposed in which the composition of the growing scale changed from oriented oxides rich in Ni and Cr to oriented oxides rich in Al. At the same time the structure changed from cubic spinels to hexagonal corundums with apparent precipitates of one phase in the matrix of the other. Eventually randomly oriented pure alpha-Al2O3 formed as the stable oxide with an abrupt transition: there was no gradual loss of orientation, no gradual compositional change or no gradual decrease in precipitate density.
Multilevel methods on unstructured grids for elliptic problems are reviewed. The advantages of these techniques are the flexible approximation of the boundaries of complicated physical domains and the ability to adapt the grid to the resolution of fine scaled structures. Multilevel methods, which include multigrid methods and domain decomposition methods, depend on the correct splitting of appropriate finite element spaces. The standard splittings used in the structured grid case cannot be directly extended to unstructured grids due to their requirement for a hierarchical grid structure. Issues related to the application of multilevel methods to unstructured grids are discussed, including how the coarse spaces and transfer operators are defined and how different types of boundary conditions are treated. An obvious way to generate a coarse mesh is to regrid the physical domain several times. Several alternatives are proposed and discussed: node nested coarse spaces, agglomerated coarse spaces and algebraically generated coarse spaces.
Direct numerical simulations (DNS) are used to examine the turbulence statistics and the radiation field generated by a high-speed turbulent boundary layer with a nominal freestream Mach number of 14 and wall temperature of 0:18 times the recovery temperature. The flow conditions fall within the range of nozzle exit conditions of the Arnold Engineering Development Center (AEDC) Hypervelocity Tunnel No. 9 facility. The streamwise domain size is approximately 200 times the boundary-layer thickness at the inlet, with a useful range of Reynolds number corresponding to Re 450 650. Consistent with previous studies of turbulent boundary layer at high Mach numbers, the weak compressibility hypothesis for turbulent boundary layers remains applicable under this flow condition and the computational results confirm the validity of both the van Driest transformation and Morkovin's scaling. The Reynolds analogy is valid at the surface; the RMS of fluctuations in the surface pressure, wall shear stress, and heat flux is 24%, 53%, and 67% of the surface mean, respectively. The magnitude and dominant frequency of pressure fluctuations are found to vary dramatically within the inner layer (z/delta 0.< or approx. 0.08 or z+ < or approx. 50). The peak of the pre-multiplied frequency spectrum of the pressure fluctuation is f(delta)/U(sub infinity) approx. 2.1 at the surface and shifts to a lower frequency of f(delta)/U(sub infinity) approx. 0.7 in the free stream where the pressure signal is predominantly acoustic. The dominant frequency of the pressure spectrum shows a significant dependence on the freestream Mach number both at the wall and in the free stream.
Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used.
In Silva et al. (2011) and Alidoost et al. (2020), the authors developed an approximation of the energy release rate field associated with a small edge or surface crack at any boundary location and with any orientation using the topological derivative. The approximation is computationally attractive because it requires only a single analysis on the non-cracked domain in contrast with conventional boundary-element and finite-element-based methods, which require a separate and costlier analysis for each crack length-location-orientation combination. Here, a shape optimization scheme for fracture-resistant structures is developed using the energy release rate approximation. In the gradient-based optimization scheme, the domain and its boundary are defined implicitly using level-set functions. The level-set functions of arbitrary geometries are constructed using Boolean operations from the level-set functions of simple primitives. This geometrical representation has the dual advantage of (i) allowing shapes to intersect and/or separate during the optimization and (ii) simplifying the computation of the shape sensitivities.
The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners.